Question

1. in triangle rst, ( mangle r = 62^circ ) and the measure of an exterior angle at t is ( 119^circ ). which is the longest side of the triangle?
2. which set of integers can not represent the respective lengths of the sides of a triangle?

(1) ( {10, 11, 12} )
(3) ( {3, 7, 10} )
(2) ( {8, 14, 18} )
(4) ( {7, 7, 13} )

1. the lengths of the sides of triangle pqr are 7, 10 and 15, respectively. find the perimeter of the triangle formed by joining the midpoints of the sides of triangle pqr.
2. find the length of the diagonal of a square whose area is 8.
3. in the accompanying diagram, ( overline{bdc} ) and ( overline{de} perp overline{aec} ). if ( mangle bda = 80 ) and ( mangle edc = 50 ), find ( mangle dac ).

Explanation:

Step1: Find ∠RTS in △RST

The exterior angle at T is supplementary to ∠RTS, so:
$m\angle RTS = 180^\circ - 119^\circ = 61^\circ$

Step2: Calculate ∠S in △RST

Sum of angles in a triangle is $180^\circ$:
$m\angle S = 180^\circ - 62^\circ - 61^\circ = 57^\circ$

Step3: Identify longest side

The longest side is opposite the largest angle ($\angle R = 62^\circ$), so it is $\overline{ST}$.

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Step1: Check triangle inequality for set (1)

Sum of two shortest sides: $10+11=21>12$, valid.

Step2: Check triangle inequality for set (2)

Sum of two shortest sides: $8+14=22>18$, valid.

Step3: Check triangle inequality for set (3)

Sum of two shortest sides: $3+7=10$, which is not greater than 10, invalid.

Step4: Check triangle inequality for set (4)

Sum of two shortest sides: $7+7=14>13$, valid.

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Step1: Apply midsegment theorem

Each midsegment is $\frac{1}{2}$ the length of the corresponding side of △PQR:
$\frac{7}{2}=3.5$, $\frac{10}{2}=5$, $\frac{15}{2}=7.5$

Step2: Calculate perimeter of new triangle

Add the midsegment lengths:
$3.5 + 5 + 7.5 = 16$

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Step1: Find side length of the square

Area $A = s^2 = 8$, so side length $s = \sqrt{8} = 2\sqrt{2}$

Step2: Calculate diagonal length

Use Pythagorean theorem for square diagonal $d$:
$d = s\sqrt{2} = 2\sqrt{2} \times \sqrt{2} = 4$

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Step1: Find ∠ADE

$\angle BDA + \angle ADE + \angle EDC = 180^\circ$, so:
$m\angle ADE = 180^\circ - 80^\circ - 50^\circ = 50^\circ$

Step2: Find ∠DAE in right △ADE

In right triangle $ADE$, $\angle AED = 90^\circ$, so:
$m\angle DAC = 180^\circ - 90^\circ - 50^\circ = 40^\circ$

Answer:

1. $\overline{ST}$
2. (3) {3, 7, 10}
3. 16
4. 4
5. $40^\circ$