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The Law of Cosines: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

The lengths of sides $a$ and $b$ are 42 and 58, respectively. The included angle $C$ measures $78^\circ$ . Find $B$ to the nearest tenth of a degree.

Answer choices for question 1 A $39.8^\circ$ B $51.0^\circ$ C $117.8^\circ$ D $62.2^\circ$
2

Two mooring posts are $64$ m and $83$ m from the same marker. The angle between the lines from the marker is $113^\circ$ . A straight cable can span at most $122$ m. Can it reach between the posts?

Answer choices for question 2 A Yes. They are about $82.7$ m apart. B Yes. They are about $104.8$ m apart. C No. The posts are about $123.0$ m apart. D No. They are about $147.0$ m apart.
3

In the coordinate setup for the cosine proof, place $C$ at the origin, $A$ at $(13,0)$ , and $B$ at $(17,6)$ . The perpendicular foot from $B$ lies beyond $A$ . What is $AB$ exactly?

Answer choices for question 3 A $\sqrt{52}$ B $\sqrt{205}$ C $\sqrt{325}$ D $10$
4

A display reports a cosine value below $-1$ when the measured lengths $14$ , $23$ , and $38$ are entered as the sides of a triangle. Which check explains the report?

Answer choices for question 4 A The negative input must be made positive to represent a length. B The longest side must appear in the denominator for any obtuse angle. C Degree mode turns a cosine below $-1$ into a valid input. D The two shorter lengths add to less than the longest, so the three sides cannot close.
5

The sides opposite $A$ , $B$ , and $C$ have lengths $28$ , $43$ , and $62$ . Which measure and angle choice correctly describe the smallest angle?

Answer choices for question 5 A $A$ is about $120.2^\circ$ . B $A$ is about $23.0^\circ$ . C $B$ is about $23.0^\circ$ . D $C$ is about $36.8^\circ$ .
6

Keep $C$ at the origin and $A$ at $(15,0)$ . Move $B$ from $(6,8)$ to $(-6,8)$ . By how much does $AB^2$ increase?

Answer choices for question 6 A $360$ B $180$ C $144$ D $0$
7

The two rigid arms of a display measure $33$ cm and $46$ cm from their common hinge. The free endpoints are $56$ cm apart. Is the opening below or above $89^\circ$ , and what is it to the nearest tenth of a degree?

Answer choices for question 7 A Above, about $91.3^\circ$ B Below, about $36.1^\circ$ C Below, about $88.7^\circ$ D Above, about $108.1^\circ$
8

The lengths of sides $a$ and $b$ of triangle $ABC$ are 35 and 50, respectively. Angle $A$ measures $42^\circ$ . How many triangles fit these data up to congruence?

Answer choices for question 8 A One, because the cosine law gives a unique answer from SAS. B Two C One, because inverse sine returns one calculator value. D Zero, because the given side opposite $A$ is shorter than $b$ .
9

A triangle with sides $r$ , $s$ , and $t$ is enlarged so that the three sides are $3r$ , $3s$ , and $3t$ . In the cosine quotient for the angle opposite the third side, what happens to the numerator and denominator?

Answer choices for question 9 A Both are multiplied by $3$ , leaving the quotient unchanged. B Both are multiplied by $9$ , leaving the quotient unchanged. C Only the numerator is multiplied by $9$ . D The numerator is multiplied by $9$ and the denominator by $3$ .
10

Two nondegenerate triangles have the same positive adjacent side lengths $a$ and $b$ . The side across from their included angle is longer in the second triangle. What does the cosine quotient show?

Answer choices for question 10 A The included angle is larger in the second triangle. B The included angle is smaller in the second triangle. C The two included angles are equal. D The comparison requires knowing whether both angles are acute.
11

A model has sides of lengths $19$ , $27$ , and $36$ . To obtain the angle facing $27$ , which cosine input is valid?

Answer choices for question 11 A $\dfrac{19^2+27^2-36^2}{2(19)(27)}$ B $\dfrac{27^2-19^2-36^2}{2(19)(36)}$ C $\dfrac{19^2+36^2-27^2}{2(19)(27)}$ D $\dfrac{19^2+36^2-27^2}{2(19)(36)}$
12

Two marked lines from a survey station end at two markers. Their lengths are known. To find the straight distance between the markers in one cosine-law step, which additional angle should be measured?

Answer choices for question 12 A An angle at one marker between its line to the station and the line to the other marker B An angle between one survey line and an unrelated reference line C The angle at the station between the lines to the markers D Either angle at a marker, chosen by which one is larger

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