Question

Find all three side lengths to the nearest hundredth and allthree angle measures to the nearest degree.B(-2,-4), C(3, 3), D(-2, 3)

Answer 1

Let's find each side first

Using the distance formula

B(-2,-4), C(3, 3),

`BC\text{ = }\sqrt[]{(3+2)^2+(3+4)^2}`
`=\sqrt[]{5^2+7^2}`
`=\sqrt[]{25+49}`
`=\sqrt[]{74}`
`=\text{ 8.60}`

C(3, 3), D(-2, 3)

`CD\text{ =}\sqrt[]{(-2-3)^2+(3-3)^2}`
`=\sqrt[]{25}`
`=5`

D(-2, 3) B(-2,-4)

`DB=\sqrt[]{(-2+2)^2+(-4-3)^2}`
`=\text{ 7}`

Now, to find the angles, use the cosine formula

BD = c = 7

BC = d = 8.6

CD = b =5

Using the formula;

`\cos \text{ B=}(c^2+d^2-b^2)/(2cd)`

substitute the values into the formula and evaluate

`\cos \text{ B=}\frac{7^2+8.6^2^{}-5^2}{2(7)(8.6)}`
`=(49+73.96-25)/(120.4)`
`=(97.96)/(120.4)`
`CosB=0.836213`

Take the cos ⁻ ' of both-side

`B=cos^(-1)(0.836213)`
`B\approx36^o`

To find angle C, we will use the cosine formula

`\text{Cos C =}(b^2+d^2-c^2)/(2bd)`

substitute the values and evaluate

`=(5^2+8.6^2-7^2)/(2(5)(8.6))`
`=(25+73.96-49)/(86)`
`=(49.96)/(86)`
`\cos C=0.58093`

Take the cos ⁻ ' of both-side

`C=\cos ^(-1)(0.58093)`

`C\approx54^o`

To find the third side

B + C + D = 180° (sum of angle in a triangle)

36 + 54 + D = 180°

90 + D = 180°

D = 180° - 90°

D = 90°