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Triangle Area with Sine: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

A ray at an obtuse angle meets the unit semicircle at height $y$ , where $0<y<1$ . Which expression is its signed horizontal projection?

Answer choices for question 1 A $\sqrt{1-y^2}$ B $-\sqrt{1+y^2}$ C $-y$ D $-\sqrt{1-y^2}$
2

Triangle P has sides of lengths $s$ and $t$ meeting at $57^\circ$ . Triangle Q has sides of lengths $3s$ and $\tfrac12t$ meeting at $123^\circ$ . How does the area of Q compare with the area of P?

Answer choices for question 2 A It is equal B It is $\tfrac32$ times as large C It is $6$ times as large D It is $\tfrac23$ times as large
3

A triangular banner has area $48.5\text{ ft}^2$ . One side at its $109^\circ$ corner measures $11.3$ ft. Find the length of the other side at that corner to the nearest tenth of a foot.

Answer choices for question 3 A $4.5$ ft B $8.1$ ft C $9.1$ ft D $18.2$ ft
4

Two rigid rods are joined at a hinge. The acute setting $36^\circ$ gives a target triangle area. Which other setting strictly between $0^\circ$ and $180^\circ$ gives exactly the same area?

Answer choices for question 4 A $144^\circ$ B $54^\circ$ C $126^\circ$ D $162^\circ$
5

In triangle $ABC$ , angle $A$ is acute and $AB$ is horizontal to the right. The length of $AC$ is $b$ , and the length of $AB$ is $c$ . When does the perpendicular from $C$ to line $AB$ land strictly beyond $B$ ?

Answer choices for question 5 A $b\sin A>c$ B $b\cos A>c$ C $b\cos A<c$ D $b>c$
6

In triangle $ABC$ , let $AB$ be the horizontal base of length $c$ and let $D$ be the perpendicular foot from $C$ . Angle $A$ is obtuse and $CD$ has length $h$ . The larger triangle $BCD$ contains $ABC$ and $ACD$ . Which subtraction gives the area of $ABC$ ?

Answer choices for question 6 A $\tfrac12(c+AD)h-\tfrac12 AD\,h$ B $\tfrac12(c+AD)h-AD\,h$ C $\tfrac12(c+AD)h$ D $\tfrac12(c-AD)h$
7

The included angle of a triangle increases from $98^\circ$ to $166^\circ$ , while the two side lengths at the angle stay fixed. Why does the area decrease?

Answer choices for question 7 A The unit point moves to the right of the vertical axis B The horizontal projection becomes positive C The unit point moves down the left half of the semicircle D The sine becomes negative
8

One triangle has two sides of lengths $14$ and $25$ meeting at angle $T$ . Another triangle has sides of lengths $10$ and $35$ meeting at $180^\circ-T$ . Which comparison is correct?

Answer choices for question 8 A The second area is greater because $35$ is the longest given side B The first area is greater because its smaller given side is longer C The areas cannot be compared without the value of $T$ D The areas are equal
9

A triangle has area $117\text{ m}^2$ and two sides of lengths $15$ m and $20$ m. Which equation determines the sine of the included angle $\theta$ ?

Answer choices for question 9 A $\sin\theta=\tfrac{39}{50}$ B $\sin\theta=\tfrac{39}{100}$ C $\sin\theta=\tfrac{50}{39}$ D $\sin\theta=\tfrac{117}{35}$
10

Two points on the upper unit semicircle have the same nonzero horizontal coordinate. What can you conclude about their angles between $0^\circ$ and $180^\circ$ ?

Answer choices for question 10 A The angles are supplementary and distinct B Their sines have opposite signs C Their angles are equal D One is acute and the other obtuse
11

A triangular sail has two edges of lengths $12.2$ m and $9.1$ m. The angle between them is $63^\circ$ . Which estimate of the area to the nearest square meter uses the correct calculation?

Answer choices for question 11 A $99\text{ m}^2$ B $25\text{ m}^2$ C $56\text{ m}^2$ D $49\text{ m}^2$
12

The area of a triangle is exactly half the product of two of its side lengths. What must be true of the angle between those two sides?

Answer choices for question 12 A It is $45^\circ$ B It is $90^\circ$ C It can be any obtuse angle D It can be any acute angle

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