Question

for # 14 - 15, clearly circle the best answer for each of the following. work must be shown in order to receive credit. 14. the legs of an isosceles right triangle are 5 inches long. what is the length of the hypotenuse of the triangle to the nearest inch? a. 7 in b. 10 in c. 25 in d. 50 in 15. triangle abc is an obtuse triangle with ( mangle c = 10^{circ} ) and ( bc>ab ). what is a possible measure of ( angle a )? a. ( 170^{circ} ) b. ( 160^{circ} ) c. ( 85^{circ} ) d. ( 10^{circ} )

Explanation:

Problem 14

Step1: Apply Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(a = b=5\) (legs of the isosceles right - triangle) and \(c\) is the hypotenuse.
Substitute \(a = 5\) and \(b = 5\) into the formula: \(c^{2}=5^{2}+5^{2}\).

Step2: Calculate \(c^{2}\)

\(c^{2}=25 + 25=50\).

Step3: Find \(c\)

\(c=\sqrt{50}\approx7.07\). Rounding \(7.07\) to the nearest inch gives \(c\approx7\).

Step1: Use triangle angle - sum property

In \(\triangle ABC\), \(m\angle A+m\angle B + m\angle C=180^{\circ}\), so \(m\angle A+m\angle B=180 - 10=170^{\circ}\).

Step2: Use the side - angle relationship

Since \(BC>AB\), then \(m\angle A>m\angle C\) (in a triangle, the larger side is opposite the larger angle). Also, since the triangle is obtuse, assume the obtuse angle is \(\angle A\).
If \(m\angle A = 85^{\circ}\), then \(m\angle B=170 - 85 = 85^{\circ}\), and the triangle would be acute.
If \(m\angle A=170^{\circ}\), then \(m\angle B = 0^{\circ}\) (not possible for a triangle).
If \(m\angle A = 160^{\circ}\), then \(m\angle B=170 - 160 = 10^{\circ}\).

Answer:

A. \(7\) in

Problem 15