TEKS: Geometry
Read an introduction to Texas Essential Knowledge and Skills charts.
TEKS  Examples  Commentary  
111.34. GEOMETRY (ONE CREDIT) (G.1) Geometric structure: knowledge and skills and performance descriptions. The student understands the structure of, and relationships within, an axiomatic system. (A) The student develops an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. (B) Through the historical development of geometric systems, the student recognizes that mathematics is developed for a variety of purposes. (C) The student compares and contrasts the structures and implications of Euclidean and nonEuclidean geometries. (G.2) The student analyzes geometric relationships in order to make and verify conjectures. (A) The student uses constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. (B) The student makes conjectures about angles, lines, polygons, circles, and threedimensional figures and determines the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. (G.3) The student applies logical reasoning to justify and prove mathematical statements. (A) The student determines the validity of a conditional statement, its converse, inverse, and contrapositive. (B) The student constructs and justifies statements about geometric figures and their properties. (C) The student demonstrates what it means to prove statements are true and find counter examples to disprove statements that are false. (D) The student uses inductive reasoning to formulate a conjecture. (E) The student uses deductive reasoning to prove a statement. (G.4) Geometric structure: The student uses a variety of representations to describe geometric relationships and solve problems. The student selects an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. (G.5) Geometric patterns: The student uses a variety of representations to describe geometric relationships and solve problems. 



(A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties. (B) The student uses numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles. 
Use a graphing calculator to explore polygons with 312 sides using the unit circle (x = cos t, y = sin t) in parametric mode by adjusting the tstep values. To draw an nsided figure, set the tstep to 360/n. Sketch the figure using graphing calculators, then transfer the figure by hand to polar paper. Determine the number of vertices, number of triangles formed by connecting one vertex to the others, sum of the angle measures, measure of each interior angle, measure of each exterior angle, sum of the measures of the exterior angles, number of diagonals, perimeter and area for a polygon with radius of 1. Generalize the pattern to write a formula for each of the explorations above for an ngon. Also, notice how the perimeter values approach the circumference of a circle and how the area values approach the area of a circle. 
Allow 12 class periods for this problem. AP* Calculus concept: Limits 

(C) The student uses properties of transformations and their compositions to make connections between mathematics and the real world in applications such as tessellations. 



(D) The student identifies and applies patterns from right triangles to solve meaningful problems, including special right triangles (454590) and (306090) and triangles whose sides are Pythagorean triples. 
If a 650 cm ladder is placed against a building at a certain angle, it just reaches a point on the building that is 520 cm above the ground. a) If the ladder is moved to reach a point 80 cm higher up, how much closer will the foot of the ladder be to the building? b) If the distance the ladder was moved inward is twice the distance it moved upward, how far is it from the wall? In right triangle ABC with right angle C, determine the measure of angles A and B using 306090 or 454590 ratios if a = 3/√2 and b = 3√6; if a = 2 and c = 4; if a = 3√2 and c = 6; etc. A boat is tied to a pier by a 25foot rope. The pier is 15 feet above the boat. If 8 feet of rope is pulled in, how many feet will the boat move forward? 
AP Calculus concept: Rates of Change  
(G.6) Dimensionality and the geometry of location: The student analyzes the relationship between threedimensional objects and related twodimensional representations and uses these representations to solve problems. (A) The student describes and draws the intersection of a given plane with various threedimensional geometric figures. (B) The student uses nets to represent and construct threedimensional objects. (C) The student uses orthographic and isometric views of threedimensional geometric figures to represent and construct threedimensional figures and solve problems. (G.7) The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. (A) The student uses one and twodimensional coordinate systems to represent points, lines, rays, line segments, and figures. (B) The student uses slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. (C) The student develops and uses formulas involving length, slope, and midpoint. (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. 



(A) The student finds areas of regular polygons, circles, and composite figures.

The rate at which Bethany's mailbox receives emails can be modeled by a continuous function. Selected values are shown in the chart below.
(a) Sketch a scatter plot of the data. (b) Draw in 4 lefthand rectangles. (c) Describe the units represented by the dimensions of the rectangle. (d) Describe the units represented by the sum of the area of the rectangles. (e) Estimate the total emails Bethany received while she was at school using the 4 lefthand rectangles. (f ) Draw in 4 righthand rectangles using a dotted line. (g) Estimate the total emails Bethany received while she was at school using the 4 righthand rectangles. (h) List the range of possible total emails using the answers to parts (e) and (g). (i) Geometrically demonstrate the error range of the 4 left and 4 righthand rectangles. (j) Create a new data table using the previous data, but showing every hour. (k) Compute the total emails Bethany received while she was at school using 8 lefthand rectangles. (l) Compute the total emails Bethany received while she was at school using 8 righthand rectangles. (m) List the range of possible total emails using the answers to parts (k) and (l). (n) List the error range and compare it to your classmates'. (They should be the same regardless of the number chosen.) (o) Estimate the total emails received using the 4 trapezoids. (Note: a trapezoid is the average of the left and righthand rectangle.) (p) Estimate the total emails Bethany received while she was at school using the 2 midpoint rectangles. 
AP Calculus Concept: Accumulation. See also AP Calculus 2000 AB2 and 99 AB3



Triangle ABC is inscribed in a semicircle centered at the origin with radius 3. Side AB of the triangle is on the xaxis and point C can be moved around the semicircle. (a) Sketch the problem situation. (b) Classify the triangle by angles. (c) Write the equation to graph the semicircle. (d) Determine the area of the triangle as a function of x. (e) List the domain for the problem situation. (f) Use a graphing calculator to determine the maximum area. Sketch the graph and justify your answers using increasing or decreasing functions and slope. (g) Determine the approximate dimensions that yield maximum area. 
AP Calculus Concept: Optimization 


Draw a circle of radius 1 inscribed in a square. Simulate the throwing of a dart to determine the probability of hitting the circle by the use of random digits. To choose a point at random in the square, choose a pair of random digits (x,y) with the appropriate limits. If the pair of random digits lies within the circle, it is considered a hit. (a) Calculate the probability of hitting inside the circle. (b) Multiply the probability by 4. What number is represented? (c) Calculate the area of the circle divided by the area of the square and multiply by 4. What number is represented? 
AP Statistics Concept: Randomization and probability 

(B) The student finds areas of sectors and arc lengths of circles using proportional reasoning. (C) The student derives, extends, and uses the Pythagorean Theorem. 
Point C is a point on a straight river. Town A is 11 miles straight across the river from C and Town B is 6 miles from that same river on the same side of the river as A. The distance from Town A to Town B is 13 miles. A pumping station is to be built along the river across from the towns at a point P to supply water to both towns. (a) Write an equation in terms of x, the distance from C to P, to express the total distance from A to P to B. (b) State the domain. (c) Use a graphing calculator to determine where the pumping station should be built in relationship to C so that the sum of the distances from the towns to the pumping station is a minimum. (d) Determine the minimum total distance. Sketch a graph and justify your answer using slopes of the curve, increasing and/or decreasing. (e) Determine the range of the distance function. (f) Using the minimum distance, determine how far the pumping station is from A and from B. 
AP Calculus concept: Optimization 

(D) The student finds surface areas and volumes of prisms, pyramids, spheres, cones, and cylinders, and composites of these figures in problem situations. 
A sheet of metal is 60 cm wide and 10 m long. It is bent along its width to form a gutter with a cross section that is an isosceles trapezoid with 120degree base angles. (a) Use 306090 ratios to express the volume of the gutter as a function of the length of one of the equal sides. (b) For what value of x is the volume of the gutter a maximum—justify algebraically. (c) List the height and base lengths for the maximum volume. (d) Determine the maximum volume. (e) If the base angle is not known, use trig ratios to express the volume of the gutter as a function of the length of one of the equal sides and the base angle, q. (f) State the domain for q. (g) Using the x value obtained in part b, verify the volume in part d using the equation in part e with unit circle values. 
AP Calculus concept: Optimization 


A cylindrical soda can is designed to hold 7π cubic inches of soda (approximately 12 ounces). The material for the top and bottom costs $0.001 per square inch. The material for the vertical surfaces cost $0.0005 per square inch. (a) Sketch the problem situation. (b) Determine the height in terms of the radius. (c) Express the cost of materials used to make the can as a function of the radius. (d) Use a graphing calculator to determine the cost of the least expensive can. (e) At the minimum cost, how much would it cost the company to produce 1,000,000 cans? (f) Determine the dimensions of the least expensive can. 
AP Calculus concept: Optimization 


Determine the volume of the solid formed by rotating the area of the region formed by:
y = (1/2)x + 1, around the (a) xaxis, (b) yaxis, (c) x = 4 
AP Calculus Concept: Volumes of Revolution. See also AP Calculus 2001 AB1, 99AB2, 98AB3, 97AB3, 96AB2 

(G.9) The student analyzes properties and describes relationships in geometric figures. (A) The student formulates and tests conjectures about the properties of parallel and perpendicular lines based on exploration and using concrete models. (B) The student formulates and tests conjectures about the properties and attributes of polygons and their component parts based on exploration and using concrete models. (C) The student formulates and tests conjectures about the properties and attributes of circles and the lines that intersect them based on exploration and using concrete models. (D) The student analyzes the characteristics of polyhedra and other threedimensional figures and their component parts based on exploration and using concrete models (G.10) The student applies the concept of congruence to justify properties of figures and solve problems. (A) The student uses congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane. (B) The student justifies and applies triangle congruence relationships. (G.11) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. (A) The student uses similarity properties and transformations to explore and justify conjectures about geometric figures. 



(B) The student uses ratios to solve problems involving similar figures. 
At a grain processing plant, the grain is falling off a conveyor and into a storage bin. The storage bin is the frustrum of a right cone with the larger base at the top. The smaller base at the bottom is closed while the grain is being poured into the bin, so the grain can be measured. When the bin is full, the contents are emptied. The top base has a radius of 3 feet, the bottom base has a radius of 2 feet. The bin has a height of 10 feet. If the height of the grain in the bin is 7.5 feet, what is the radius of the grain? 
AP Calculus concept: Rates of Change 

(C) The student develops, applies, and justifies triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods. 
Determine the coordinates on the unit circle for multiples of 30 and 45 degrees for measures from 0 to 360 degrees using 306090 and 454590 ratios. Introduce the concept of radian measure in terms of arc length for 0 to 2π radians. A tightrope is stretched 30 feet above the ground between the Jay and the Tee buildings, which are 50 feet apart. A spotlight, at the top of the Jay building 70 feet above the tightrope, illuminates a tightrope walker. She is walking at a constant rate of 2 feet per second from the Jay building to the Tee building. (a) Sketch the problem situation. (b) Letting the distance she has walked on the tightrope be x and the letting the distance her shadow has moved along the ground be y, determine y in terms of x using a similar triangle. (c) How far from the Jay building is the tightrope walker when her shadow reaches the Tee building? (d) Where does her shadow move when it reaches the Tee building? (e) Determine the length of her shadow on the wall in terms of x using a new set of similar triangles. 
This should be done as two separate lessons and emphasized continually throughout the course. Students should be able to produce the unit circle with measurements in degrees and radians by memory. AP Calculus Concepts: Related Rates of Change Adapted from AP Calculus 1991 AB6 

(4) The student describes the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and applies this idea in solving problems. 
Suppose cylinder A has twice the radius but half the height of cylinder B. How do the cylinder's lateral areas, total area and volumes compare? If the length and width of a rectangular solid are each decreased by 20%, by what percent must the height be increased for the volume to remain unchanged? 
