Trigonometry Question

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The questions below relate to the following proof.

For

$Delta ABC$ , where

$m ang BAC > 90°$ , let the intersection of the altitude from vertex

$B$ and the extension of

$bar{AC}$ be point

$D$ (not labeled). Let

$theta = m ang BAC$ and

$alpha = m ang BAD$ . Prove that

$BC^2 = AC^2 + AB^2 - 2 AC \ AB cos(theta)$ .

Obtuse Triangle Height v2

$" Statement "$	$" Reason "$
$1. bar{BD} " is an altitude"$	$1. "Given"$
$2. bar{BD} _\|_ bar{DC}$	$2. "Definition of an altitude"$
$3. ang BDC " is a right angle"$	$3. "Definition of perpendicular lines"$
$4. Delta BDA, \ Delta BDC " are right triangles"$	$4. "Definition of right triangles"$
$5.$	$5. "C""osine ratio in a right triangle"$
$6. AB cos(alpha) = AD$	$6. "Multiplication Property of Equality"$
$7.$	$7. "S""ine ratio in a right triangle"$
$8. AB sin(alpha) = BD$	$8. "Multiplication Property of Equality"$
$9. BC^2 = BD^2 + CD^2$	$9. ""$
$10.$	$10. "Segment Addition Postulate"$
$11. BC^2 = BD^2 + (AD + AC)^2$	$11. "Substitution Property of Equality"$
$12. BC^2 = (AB sin(alpha))^2 + (ABcos(alpha) + AC)^2$	$12. ""$
$13. BC^2 = AB^2 sin^2(alpha) + (ABcos(alpha) + AC)^2$	$13. "Distributive Property of Exponents"$
$14. BC^2 = AB^2 sin^2(alpha) + AB^2cos^2(alpha) +$ $\ \ \ 2 AB \ AC cos(alpha) + AC^2$	$14. "Distributive Property"$
$15. BC^2 = AB^2 (sin^2(alpha) + cos^2(alpha)) +$ $\ \ \ 2 AB \ AC cos(alpha) + AC^2$	$15. "Distributive Property"$
$16. BC^2 = AB^2 + 2 AB \ AC cos(alpha) + AC^2$	$16. ""$
$17. BC^2 = AB^2 + AC^2 + 2 AB \ AC cos(alpha)$	$17. "Commutative Property of Addition"$
$18. alpha + theta = 180°$	$18. ""$
$19. alpha = 180° - theta$	$19. "Subtraction Property of Equality"$
$20. BC^2 = AB^2 + AC^2 + 2 AB \ AC cos(180° - theta)$	$20. "Substitution Property of Equality"$
$21. BC^2 = AB^2 + AC^2 - 2 AB \ AC cos(theta)$	$21. "Trigonometric Identity"$

Grade 11 Trigonometry CCSS: HSG-SRT.D.10

What is the missing reason in step 16?

Pythagorean Identity
Pythagorean Theorem
Algebra (subtraction)
Subtraction Property of Equality

Question Info

Trigonometry Question

Grade 11 Trigonometry CCSS: HSG-SRT.D.10