Question

pythagorean theorem converse
given a triangle with sides a, b, and c:

• if _________________, then the triangle is __________.
• if _________________, then the triangle is __________.
• if _________________, then the triangle is __________.

directions: given the side lengths, determine whether the triangle is acute, right, obtuse, or not a triangle.

1. 15, 16, 21

☐ not a δ
☐ acute
☐ right
☐ obtuse

1. 20, 23, 41

☐ not a δ
☐ acute
☐ right
☐ obtuse

1. 10, 24, 26

☐ not a δ
☐ acute
☐ right
☐ obtuse

1. 6, 13, 20

☐ not a δ
☐ acute
☐ right
☐ obtuse

1. 3, 16, 17

☐ not a δ
☐ acute
☐ right
☐ obtuse

1. 24, 29, 32

☐ not a δ
☐ acute
☐ right
☐ obtuse

Explanation:

First, we define the converse rules for the Pythagorean theorem (let $c$ be the longest side of the triangle):

Step1: Right triangle condition

If $a^2 + b^2 = c^2$, then the triangle is right.

Step2: Acute triangle condition

If $a^2 + b^2 > c^2$, then the triangle is acute.

Step3: Obtuse triangle condition

If $a^2 + b^2 < c^2$, then the triangle is obtuse.
Also, first verify the triangle inequality: sum of two shorter sides > longest side.

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For 11. 15, 16, 21

Step1: Identify longest side, compute squares

Longest side $c=21$. Calculate $15^2+16^2=225+256=481$, $21^2=441$.

Step2: Compare values

$481 > 441$, so $a^2+b^2 > c^2$.

For 12. 20, 23, 41

Step1: Check triangle inequality

$20+23=43$, $43 < 41$ is false? No, $20+23=43 > 41$. Now compute squares: $20^2+23^2=400+529=929$, $41^2=1681$.

Step2: Compare values

$929 < 1681$, so $a^2+b^2 < c^2$.

For 13. 10, 24, 26

Step1: Identify longest side, compute squares

Longest side $c=26$. Calculate $10^2+24^2=100+576=676$, $26^2=676$.

Step2: Compare values

$676 = 676$, so $a^2+b^2 = c^2$.

For 14. 6, 13, 20

Step1: Check triangle inequality

$6+13=19$, $19 < 20$, so triangle inequality fails.

For 15. 3, 16, 17

Step1: Identify longest side, compute squares

Longest side $c=17$. Calculate $3^2+16^2=9+256=265$, $17^2=289$.

Step2: Compare values

$265 < 289$, so $a^2+b^2 < c^2$.

For 16. 24, 29, 32

Step1: Identify longest side, compute squares

Longest side $c=32$. Calculate $24^2+29^2=576+841=1417$, $32^2=1024$.

Step2: Compare values

$1417 > 1024$, so $a^2+b^2 > c^2$.

Answer:

Pythagorean Theorem Converse Rules:

• If $\boldsymbol{a^2 + b^2 = c^2}$, then the triangle is $\boldsymbol{\text{right}}$.
• If $\boldsymbol{a^2 + b^2 > c^2}$, then the triangle is $\boldsymbol{\text{acute}}$.
• If $\boldsymbol{a^2 + b^2 < c^2}$, then the triangle is $\boldsymbol{\text{obtuse}}$.

Side Length Classifications:

1. $\boldsymbol{\text{Acute}}$
2. $\boldsymbol{\text{Obtuse}}$
3. $\boldsymbol{\text{Right}}$
4. $\boldsymbol{\text{Not a }\Delta}$
5. $\boldsymbol{\text{Obtuse}}$
6. $\boldsymbol{\text{Acute}}$