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This question group is public and is used in 3 tests.

Author: nsharp1
No. Questions: 10
Created: Feb 17, 2021
Last Modified: 4 years ago

Prove Law of Sines, Acute

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The questions below concern the following proof.

Let

$Delta ABC$ be an acute triangle. Let

$A$ represent

$m ang A$ ,

$B$ represent

$m ang B$ , and

$C$ represent

$m ang C$ . Prove the law of sines,

$(BC) / sin(A) = (AC) / sin(B) = (AB) / sin(C)$ .

Acute Triangle ABC v2

Acute Triangle ABC v2

$" Statement "$	$" Reason "$
$1. "Construct altitude" \ \bar{BD} \ "such that" \ D \ "lies on" \ bar{AC}$	$1. ""$
$2. \bar{BD} _\|_ bar{AC}$	$2. "Definition of an altitude"$
$3. ang ADB, \ ang BDC " are right angles"$	$3. "Definition of perpendicular lines"$
$4. Delta ADB, \ Delta BDC " are right triangles"$	$4. "Definition of right triangles"$
$5.$	$5. "S""ine ratio in a right triangle"$
$6. AB sin(A) = BD$	$6. "Multiplication Property of Equality"$
$7.$	$7. "S""ine ratio in a right triangle"$
$8. BC sin(C) = BD$	$8. "Multiplication Property of Equality"$
$9. AB sin(A) = BC sin(C)$	$9. ""$
$10. AB = (BC sin(C))/sin(A)$	$10. "Division Property of Equality"$
$11. (AB)/sin(C) = (BC)/sin(A)$	$11. "Division Property of Equality"$
$12. "Construct altitude" \ \bar{AE} \ "such that" \ E \ "lies on" \ \bar[BC}$	$12. ""$
$13. \bar{AE} _\|_ \bar{BC}$	$13. "Definition of an altitude"$
$14. ang AEB, \ ang AEC " are right angles"$	$14. "Definition of perpendicular lines"$
$15. Delta AEB, \ Delta AEC " are right triangles"$	$15. "Definition of right triangles"$
$16.$	$16. "S""ine ratio in a right triangle"$
$17. AC sin(C) = AE$	$17. "Multiplication Property of Equality"$
$18.$	$18. "S""ine ratio in a right triangle"$
$19. AB sin(B) = AE$	$19. "Multiplication Property of Equality"$
$20.$	$20. "Transitive Property of Equality"$
$21. (AB sin(B))/sin(C) = AC$	$21. "Division Property of Equality"$
$22. (AB)/sin(C) = (AC)/sin(B)$	$22. "Division Property of Equality"$
$23. (BC)/sin(A) = (AC)/sin(B)$	$23. ""$
$24. (BC)/sin(A) = (AC)/sin(B) = (AB)/sin(C)$	$24. "Combined results"$

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

A.

What is the missing reason in step 1?

Given
All acute triangles have altitudes from each vertex that intersect the opposite side
Perpendicular Bisector Theorem (for triangles)
All acute triangles have an orthocenter that lies within the triangle

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

B.

What is the missing statement in step 5?

$sin(A) = (BD)/(AB)$
$sin(A) = (BD)/(AD)$
$sin(A) = (BC)/(AC)$
$sin(A) = (AD)/(AB)$

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

C.

What is the missing statement in step 7?

$sin(C) = (BD)/(AB)$
$sin(C) = (CS)/(BC)$
$sin(C) = (BC)/(AC)$
$sin(C) = (BD)/(BC)$

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

D.

What is the missing statement in step 9?

Transitive Property of Equality
Law of sines
Multiplication Property of Equality
Trigonometric Identity

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

E.

What is the missing reason in step 12?

Each of the three altitudes of an acute triangle intersects the side opposite the vertex through which the altitude passes
The orthocenter of all acute triangles lies within the triangle
Perpendicular Bisector Theorem (for triangles)
All triangles have three altitudes

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

F.

What is the missing statement in step 16?

$sin(C) = (AB)/(AC)$
$sin(C) = (CE)/(AC)$
$sin(C) = (AE)/(AC)$
$sin(C) = (AE)/(CE)$

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

G.

What is the missing statement in step 18?

$sin(B) = (CD)/(BC)$
$sin(B) = (BE)/(AB)$
$sin(B) = (AB)/(AC)$
$sin(B) = (AE)/(AB)$

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

H.

What is the missing statement in step 20?

$AB sin(B) = AC sin(C)$
$AE = AE$
$sin(C) = sin(B)$
$AC = AB$

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

I.

What is the missing reason in step 23?

Previous result
Transitive Property of Equality
Multiplication Property of Equality
Law of sines

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

J.

The proof given applies to which type of triangles?

All triangles.
Acute triangles.
Non-right triangles.
Scalene triangles.