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Solving Right Triangles: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

A right triangle has opposite leg 5.5 cm and adjacent leg 12.3 cm relative to theta. Which expression correctly finds its hypotenuse without an intermediate rounded angle?

Answer choices for question 1 A $\frac{12.3}{\cos(\tan^{-1}(\frac{5.5}{12.3}))}$ B $12.3\sin(\tan^{-1}(\frac{5.5}{12.3}))$ C $12.3\cos(\tan^{-1}(\frac{5.5}{12.3}))$ D $\frac{12.3}{\cos(\tan^{-1}(\frac{12.3}{5.5}))}$
2

Triangle ABC is right at C, with BC of 8 units and CA of 15 units. Which expression finds angle B directly from the two legs?

Answer choices for question 2 A $\cos^{-1}(15/17)$ B $\tan^{-1}(17/15)$ C $\sin^{-1}(15/8)$ D $\tan^{-1}(15/8)$
3

A record calls a 6-unit side the hypotenuse and an 8-unit side the leg opposite an acute angle. What is the appropriate conclusion?

Answer choices for question 3 A Reverse 8/6 to 6/8 and accept the record unchanged. B Use inverse tangent on 8/6 to make the hypotenuse condition valid. C The data are inconsistent because the supposed leg exceeds the hypotenuse. D Only a calculator setting prevents this from being a valid right triangle.
4

A right triangle has an acute angle of 42 degrees and perimeter 50 cm. Which expression gives its hypotenuse?

Answer choices for question 4 A $50/(\sin42^\circ+\cos42^\circ)$ B $50/(1+\sin42^\circ+\cos42^\circ)$ C $50(1+\sin42^\circ+\cos42^\circ)$ D $50/(1+\sin42^\circ)$
5

Which pair of ratios can describe the same acute angle in a nondegenerate right triangle?

Answer choices for question 5 A
$\sin\theta=3/5$

$\cos\theta=3/5$
B
$\tan\theta=3/4$

$\cos\theta=5/4$
C
$\sin\theta=3/5$

$\cos\theta=4/5$
D
$\sin\theta=5/3$

$\cos\theta=4/3$
6

An acute angle in a right triangle has $\cos\theta=0.08$ . Its adjacent leg is 2.5 m. Which expression finds the hypotenuse directly, without first finding theta?

Answer choices for question 6 A $2.5(0.08)$ B $2.5/0.08$ C $0.08/2.5$ D $\cos^{-1}(0.08)/2.5$
7

A right triangle has opposite leg 16 units and adjacent leg 7 units relative to theta. Which direct inverse expression has valid side roles and input?

Answer choices for question 7 A $\tan^{-1}(16/7)$ B $\sin^{-1}(16/7)$ C $\cos^{-1}(16/7)$ D $\tan^{-1}(7/16)$
8

Exact side data give $\sin\theta=0.9995$ for an acute angle. Which procedure preserves the original valid angle calculation?

Answer choices for question 8 A Round the ratio to 1.00 before taking $\sin^{-1}$ . B Use $1/0.9995$ as the angle. C Evaluate $\sin(0.9995^\circ)$ . D Evaluate $\sin^{-1}(0.9995)$ , then round the angle as requested.
9

A nondegenerate right triangle is said to have an acute angle theta with $\cos\theta=-0.2$ . What is the correct response?

Answer choices for question 9 A Take the absolute value and claim the original data are satisfied. B Reject the acute-angle record because a positive adjacent leg divided by a positive hypotenuse cannot be negative. C Treat the negative sign as a negative side length. D Change calculator mode and accept any displayed angle.
10

Which information uniquely determines all side lengths of a nondegenerate right triangle?

Answer choices for question 10 A One acute angle and the positive perimeter. B Both acute angles, with no lengths. C One leg length, with no acute angle. D The ratio of the two legs, with no length.
11

A right triangle has hypotenuse 18 cm and an acute angle of 47 degrees. Find its opposite leg to the nearest hundredth of a centimeter.

Answer choices for question 11 A $13.2$ cm B $12.28$ cm C $24.61$ cm D $13.16$ cm
12

A right triangle has exact legs of 11 and 17 units. A solver reports its hypotenuse as 20.2 units and its angle opposite 11 as 32.9 degrees, each rounded to the nearest tenth. Which assessment is correct?

Answer choices for question 12 A The hypotenuse is wrong because its rounded square is not exactly 410. B The angle is wrong because inverse tangent must be replaced by a reciprocal. C Both reported values are consistent with the exact data after the stated rounding. D Neither result can be checked from two legs alone.

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