Question

a triangle has legs of 9 cm and 12 cm. what is the length of the hypotenuse?
a. 15 cm
b. 20 cm
c. 13 cm
d. 17 cm

two triangles are similar, and one triangle has sides 8 cm, 12 cm, and 16 cm. the corresponding sides of the other triangle are proportional, with the shortest side measuring 6 cm. what is the length of the longest side of the second triangle?
a. 12 cm
b. 20 cm
c. 18 cm
d. 24 cm

in a right triangle, the hypotenuse is 26 cm, and one leg is 10 cm. what is the length of the other leg?
a. 24 cm
b. 20 cm
c. 18 cm
d. 22 cm

a surveyor measures a triangular plot of land and finds two sides are 45 meters and 60 meters. the hypotenuse of the triangle is 75 meters. which theorem justifies this relationship?
a. pythagorean theorem
b. triangle proportionality theorem
c. sas similarity criterion
d. aa similarity criterion

a ladder leaning against a building creates a right triangle with the ground. the ladder is 20 ft long, and the base of the ladder is 16 ft from the building. how high does the ladder reach on the building?
a. 15 ft
b. 11 ft
c. 12 ft
d. 10 ft

which postulate or theorem proves two triangles are congruent if two angles and the included side are equal?
a. sss
b. asa
c. aas
d. sas

Explanation:

Step1: Use Pythagorean theorem

For a right - triangle with legs \(a = 9\) and \(b=12\), and hypotenuse \(c\), the Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\). Substitute \(a = 9\) and \(b = 12\) into the formula: \(9^{2}+12^{2}=c^{2}\).

Step2: Calculate \(a^{2}+b^{2}\)

\(9^{2}=81\) and \(12^{2}=144\). Then \(81 + 144=c^{2}\), so \(c^{2}=225\).

Step3: Solve for \(c\)

Take the square root of both sides: \(c=\sqrt{225}=15\).

Step1: Find the scale factor

The first triangle has sides \(8\), \(12\), \(16\) (shortest side \(8\)). The second triangle has shortest side \(6\). The scale factor \(k=\frac{6}{8}=\frac{3}{4}\).

Step2: Find the longest side of the second triangle

The longest side of the first triangle is \(16\). Multiply by the scale factor: \(16\times\frac{3}{4}=12\).

Step1: Use Pythagorean theorem

Let the hypotenuse \(c = 26\), one leg \(a = 10\), and the other leg \(b\). By \(a^{2}+b^{2}=c^{2}\), we have \(b^{2}=c^{2}-a^{2}\).

Step2: Substitute values

\(b^{2}=26^{2}-10^{2}\). Calculate \(26^{2}=676\) and \(10^{2}=100\). Then \(b^{2}=676 - 100=576\).

Step3: Solve for \(b\)

\(b=\sqrt{576}=24\).

Step1: Recall Pythagorean theorem

The Pythagorean theorem states that for a right - triangle \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. Here \(45^{2}+60^{2}=2025 + 3600=5625\) and \(75^{2}=5625\).

Answer:

a. 15 cm