Question

determine whether each set of numbers can be measure of the sides of a triangle. if so, classify the triangle as acute, obtuse, or right. justify your answer.

1. 14, 48, 50
2. 16, 30, 34
3. 5, 5, 9
4. 4, 5, 7
5. 18, 24, 30
6. 10, 24, 26

Explanation:

For each set, first verify the triangle inequality (sum of any two sides > third side), then use the converse of the Pythagorean theorem: let $c$ be the longest side.

• If $a^2 + b^2 = c^2$: right triangle
• If $a^2 + b^2 > c^2$: acute triangle
• If $a^2 + b^2 < c^2$: obtuse triangle

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1. Sides: 14, 48, 50

Step1: Verify triangle inequality

$14+48=62>50$, $14+50=64>48$, $48+50=98>14$

Step2: Test Pythagorean converse

$14^2 + 48^2 = 196 + 2304 = 2500$; $50^2=2500$
$14^2 + 48^2 = 50^2$

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2. Sides: 16, 30, 34

Step1: Verify triangle inequality

$16+30=46>34$, $16+34=50>30$, $30+34=64>16$

Step2: Test Pythagorean converse

$16^2 + 30^2 = 256 + 900 = 1156$; $34^2=1156$
$16^2 + 30^2 = 34^2$

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3. Sides: 5, 5, 9

Step1: Verify triangle inequality

$5+5=10>9$, $5+9=14>5$, $5+9=14>5$

Step2: Test Pythagorean converse

$5^2 + 5^2 = 25 + 25 = 50$; $9^2=81$
$5^2 + 5^2 < 9^2$

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4. Sides: 4, 5, 7

Step1: Verify triangle inequality

$4+5=9>7$, $4+7=11>5$, $5+7=12>4$

Step2: Test Pythagorean converse

$4^2 + 5^2 = 16 + 25 = 41$; $7^2=49$
$4^2 + 5^2 < 7^2$

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5. Sides: 18, 24, 30

Step1: Verify triangle inequality

$18+24=42>30$, $18+30=48>24$, $24+30=54>18$

Step2: Test Pythagorean converse

$18^2 + 24^2 = 324 + 576 = 900$; $30^2=900$
$18^2 + 24^2 = 30^2$

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6. Sides: 10, 24, 26

Step1: Verify triangle inequality

$10+24=34>26$, $10+26=36>24$, $24+26=50>10$

Step2: Test Pythagorean converse

$10^2 + 24^2 = 100 + 576 = 676$; $26^2=676$
$10^2 + 24^2 = 26^2$

Answer:

1. Yes, right triangle
2. Yes, right triangle
3. Yes, obtuse triangle
4. Yes, obtuse triangle
5. Yes, right triangle
6. Yes, right triangle