Common Core Standard HSG-SRT.D.9 Questions

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Grade 11 Trigonometry CCSS: HSG-SRT.D.9

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What is the missing statement in step 9?

$sin(alpha) = (AD)/(AB)$
$sin(alpha) = (AD)/(BD)$
$sin(alpha) = (BD)/(AD)$
$sin(alpha)= (BD)/(AB)$

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

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Is the sine formula for the area of a triangle valid for the acute angles in an obtuse triangle?

Yes, but the derivation is very complex.
Yes, and it can be derived the same way it is in an acute triangle.
Yes, since an obtuse triangle, if divided into two smaller triangles, is ALWAYS two acute triangles, and the formula can then be derived for acute triangles.
No, a different formula is used.

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

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Can the sine formula for the area of a triangle be applied to

$Delta ABC$ using sides

$bar{AB}$ and

$bar{AC} ?$

Yes, since $sin(90°)=1$ , $A_{Delta ABC} = 1/2 AB \ AC sin(90°) = 1/2 AB \ AC$ , which is also the standard formula for the area of a triangle as the legs of $Delta ABC$ represent the base and height of the triangle.
Yes, since $sin(90°) = 0$ , the sine term in the formula is ignored and therefore $A_{Delta ABC} = 1/2 AB \ AC$ , which is also the standard formula for the area of a triangle as the legs of $Delta ABC$ represent the base and height of the triangle.
No, since the angle between those sides may not be specified for some right triangles.
No, since the the sine of a right angle does not exist.

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

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Can the sine formula for the area of a triangle be derived for the acute angles of

$Delta ABC ?$

Yes, although the derivation is much more complicated since the legs of the right triangle are also altitudes.
Yes, simply using the standard formula for the area of a triangle and the sine ratio for either angle.
No, it cannot, but the formula is still valid.
No, it cannot, and the formula is not valid for acute angles of right triangles.

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

For acute triangle ABC, as pictured, derive the formula

$A_{Delta ABC} = 1/2 AB \ BC sin(m ang ABC)$ using the sine addition formula,

$sin(x+y) = sin(x)cos(y) + sin(y)cos(x)$ . Let the intersection of the altitude from vertex

$B$ and the opposite side

$bar{AC}$ be point

$D$ . Also, let

$m ang ABD = alpha$ and

$m ang DBC = theta$ .

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

Given that

$AC = 5 \ "units"$ ,

$AB = 9 \ "units"$ and

$m ang A = 110°$ , what is the area of

$Delta ABC ?$ Round the answer to one decimal place.

$4.7 \ "units"^2$
$8.5 \ "units"^2$
$21.1 \ "units"^2$
$42.3 \ "units"^2$

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

$Delta ABC$ ,

$AB = 4.6 \ "units"$ and

$BC = 3.8 \ "untis"$ . If

$m ang B = 56°$ , what is the area of the triangle?

$3.6 \ "units"^2$
$7.2 \ "units"^2$
$14.5 \ "units"^2$
$17.5 \ "units"^2$

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

$Delta ABC$ ,

$AB = 12 \ "units"$ ,

$BC = 4.3 \ "units"$ , and

$m ang B = 100°$ . What is the area of

$Delta ABC ?$

$25.4 \ "units"^2$
$50.8 \ "units"^2$
$51.6 \ "units"^2$
$101.6 \ "units"^2$

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

$Delta ABC$ ,

$AB = 11.2 \ "units"$ ,

$AC = 3.8 \ "units"$ , and

$m ang A = 79°$ . What is the length of the altitude that intersects vertex

$B ?$ Round the answer to one decimal place.

$3.7 \ "units"$
$5.5 \ "units"$
$5.9 \ "units"$
$11.0 \ "units"$

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

$Delta ABC$ , let the intersection of the altitude from vertex

$B$ and side

$bar{AC}$ be point

$D$ (not labeled). Also,

$BC = 20.1 \ "units"$ ,

$AB = 18.2 \ "units"$ , and

$m ang ABC = 72.2°$ . If

$BD:AC = 17:25$ , what is the length of

$bar{BD}$ , rounded to one decimal place?

10.9 units
15.4 units
16.4 units
22.6 units

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

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What is the missing reason in step 3?

$bar{BD}$ and $bar{AC}$ are not parallel, so they must be perpendicular
Triangle Bisector Theorem
Perpendicular Bisector Theorem
Definition of an altitude

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

$Delta ABC$ ,

$AB = 3.1 \ "units"$ and

$BC = 4.0 \ "units"$ . If the area of the triangle is

$5.7 \ "units"^2$ , what is the measure of

$ang B ?$ Round the answer to one decimal place.

6.2°
13.3°
27.4°
66.8°

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

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What is the missing statement in step 6?

$sin(theta) = (AD)/(AB)$
$sin(theta) = (BD)/(AB)$
$sin(theta) = (AD)/(BD)$
$sin(theta) = (BD)/(AD)$

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

$Delta ABC$ , let the intersection of the altitude from

$B$ and the side

$bar{AC}$ be point

$D$ (not labeled). Also,

$AB = 1.2 \ "units"$ ,

$AC = 3.9 \ "units"$ , and the area of

$Delta ABC$ is

$0.88 \ "units"^2$ . If the area of

$Delta BDC$ is three times as great as the area of

$Delta ABD$ , what is the length of

$bar{AD}$ , rounded to two decimal places?

$0.98 \ "units"$
$1.30 \ "units"$
$1.95 \ "units"$
Not enough information.

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

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What is the missing reason in step 13?

Given
Area of a triangle
Substitution Property of Equality
Multiplication Property of Equality

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

$Delta ABC$ ,

$AB = 32.5 \ "units"$ ,

$m ang A = 27.2°$ , and the area of the triangle is

$281.5 \ "units"^2$ . What is the length of

$bar{AC}$ , rounded to one decimal place?

$37.9 \ "units"$
$18.9 \ "units"$
$17.3 \ "units"$
$9.5 \ "units"$

Grade 11 Trigonometry CCSS: HSG-SRT.D.9

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Does it matter that the formula specified

$ang A$ and not one of the other angles in the triangle?

No. Any angle in an acute triangle can be chosen, with an altitude not intersecting the vertex of that angle, and the method of derivation will be the same.
No. Although the derivation will be different depending on the angle chosen (whether its measure is the largest, middle, or smallest of the triangle).
Yes. This formula is only valid for the two angles with the smaller measures in an acute triangle.
Yes. This formula cannot be derived for $ang B$ in any manner, since the altitude intersects vertex $B$ .