Soccer Season Report: Proof Behind the Graphics

Explain why the Pythagorean Theorem and its converse work.

The Standings Page: An Area Proof

The soccer league's season report includes a geometry feature beside its standings. Use the model to explain the proof.

The report designer places four congruent right triangles inside an outer square. Each triangle has legs a and b and hypotenuse c. The triangles do not overlap. Their hypotenuses surround a central region, and each outer side has length a + b.

Complete the model by connecting S to P. The outer square has corners (0, 0), (10, 0), (10, 10), and (0, 10). The four corner triangles have a = 4 and b = 6. The central region is PQRS. The same arrangement can be made for any positive leg lengths a and b.

Use the grid model. What is the total area of the four corner triangles?

Total triangle area: square units

In the general arrangement, let α and β be the two acute angles of each right triangle. Why is every interior angle of the central region 90°?

A. It equals α + β + 90°, which is 90°.B. It equals 360° − α − β, which is 90°.C. It equals 90° − α − β, which is 90°.D. It equals 180° − α − β, which is 90°.

The angle argument proves that all four central angles are right angles. Which additional fact proves that the central region is a square with area c²?

A. Each central side is a triangle's shorter leg, so each has length a.B. Each central side is an outer side, so each has length a + b.C. Each central side is a triangle's longer leg, so each has length b.D. Each central side is a triangle's hypotenuse, so each has length c.

Which equation correctly accounts for all the area in the general arrangement?

A. (a + b)² = 2ab + c²B. (a + b)² = ab + c²C. (a + b)² = 4ab + c²D. a² + b² = 2ab + c²

Expanding the correct area equation gives a² + 2ab + b² = 2ab + c². Which step finishes the proof?

A. Divide every term by 2ab to obtain a² + b² = c².B. Replace (a + b)² with a² + b² to obtain a² + b² = c².C. Subtract 2ab from both sides to obtain a² + b² = c².D. Subtract a² + b² from both sides to obtain a² + b² = c².

Player-Stat Cards: Proving the Converse

The report uses triangular frames for player-stat cards. Follow the proof that a side-length equation guarantees a right angle.

A triangular frame has positive side lengths a, b, and c, with c the longest side. Suppose a² + b² = c². To prove the converse, construct a separate right triangle with legs a and b. Call its hypotenuse h.

Why is it valid to write h² = a² + b² for the separately constructed triangle?

A. The theorem applies because the constructed triangle has a right angle.B. The SSS rule says any two triangles have equal squared side lengths.C. The converse applies because the original frame is already known to be right.D. The triangle inequality says the square of any side equals that sum.

The constructed triangle satisfies h² = a² + b², and the frame satisfies c² = a² + b². Why must h = c?

A. Their squares are equal, and both lengths are positive.B. Their squares are equal, so one length must be negative.C. Their products with b are equal, so both lengths equal a.D. Their sums with a are equal, so both lengths equal b.

Match each remaining proof step to its reason.

An editor writes: “The frame satisfies a² + b² = c². Apply the Pythagorean Theorem directly to conclude that the frame is right.” What is the gap in this argument?

A. It should use the shortest side in place of c in the equation.B. It uses a theorem requiring a right angle before establishing that angle.C. It assumes that every triangle can be divided into two equal triangles.D. It should add the side lengths instead of adding their squares.

Review the Season Report

Use proof ideas to review the standings and player-stat graphics. All listed measurements are exact.

10.

The standings panel has a triangular corner graphic with side lengths 8 cm, 15 cm, and 17 cm. Which statement justifies labeling the angle opposite the 17 cm side as a right angle?

A. Since all three lengths are whole numbers, one angle must be a right angle.B. Since 8² + 15² = 17², the converse proves that the angle is right.C. Since 8 + 15 > 17, the triangle inequality proves that the angle is right.D. Since 17 is the longest side, its opposite angle must be a right angle.

11.

A top-scorer card uses a triangular frame with sides 7 cm, 10 cm, and 12 cm. The designer claims it has a right angle. Which statement correctly reviews the claim?

A. The claim is false because a right triangle needs two equal sides.B. The claim is proved because 7 + 10 is greater than 12.C. The claim is false because 7² + 10² is 149, not 12².D. The claim is proved because 12 is longer than both other sides.

12.

Classify each statement from the report review by the direction of reasoning it needs.

13.

A player-stat frame has side lengths 12 cm, 35 cm, and 37 cm. Lena constructs a separate right triangle with legs 12 cm and 35 cm. Explain how this construction proves that the original frame has a right angle. Include the length of the constructed hypotenuse and the congruence reason.