### Problem Statement:In the given right-angled triangle ΔABC, angle ∠C is 90°. The vertices are labeled as follows:- A at the left-most point,- B at the top-right point, and- C at the bottom-right point where the right angle is formed.The sides of the triangle are designated as:- Side opposite to angle A (between B and C) is 'a',- Side adjacent to angle A (between A and C) is 'b',- Hypotenuse (between A and B) is 'c'.Given:- Length of the hypotenuse, c = 15 units- Measure of angle ∠B = 78°Question:- Find the length of side 'a'.### Diagram:Here we have a right-angled triangle with:- Right angle at C (∠C = 90°)- Hypotenuse (c) connecting points A and B- Vertical side (a) connecting points B and C- Horizontal side (b) connecting points A and C.### Solution:To find the length of side 'a', we utilize trigonometric relationships in the right-angled triangle. Given that:- \( c = 15 \)- \( \angle B = 78° \)To find side 'a', observe that in triangle ΔABC:\[\sin(\angle B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c}\]Thus,\[\sin(78°) = \frac{a}{15}\]We can solve for 'a':\[a = 15 \times \sin(78°)\]Using a calculator to find \( \sin(78°) \):\[\sin(78°) \approx 0.9781\]Therefore:\[a \approx 15 \times 0.9781 \approx 14.67\]So, the length of side 'a' is approximately 14.67 units.

In a right angle triangle, the longest side is called hypotenuse, the side opposite to the angle x…

Answered: B C. a A b c = 15 and mZB= 78°. Find a.

B C. a A b c = 15 and mZB= 78°. Find a.

1st Edition

ISBN:9780547587776

Author:HOLT MCDOUGAL

Publisher:HOLT MCDOUGAL

Chapter12: Angle Relationships And Transformations

Section12.5: Reflections And Symmetry

Problem 20E

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