Question

a) fill in the table.

triangle	$a^2$	$b^2$	$a^2 + b^2$	$c^2$	$a^2 + b^2 \\_ c^2$ ($<, >, =$)	acute/obtuse/right triangle
(triangle with sides 10 ft, 17 ft, 18 ft)
(triangle with sides 20 yd, 11 yd, 13 yd)

b) fill in the table.

sides of the triangle	$a^2$	$b^2$	$a^2 + b^2$	$c^2$	$a^2 + b^2 \\_ c^2$ ($<, >, =$)	acute/obtuse/right triangle
$a = 15$ yd; $b = 8$ yd; $c = 17$ yd
$a = 9$ in; $b = 18$ in; $c = 19$ in
$a = 5$ ft; $b = 7$ ft; $c = 11$ ft

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Answer:

Part A: Triangles with Given Sides (by Length)

Triangle 1: \( BC = 12 \) in, \( AC = 5 \) in, \( AB = 13 \) in

Let \( a = 5 \), \( b = 12 \), \( c = 13 \) (hypotenuse, longest side).

• \( a^2 = 5^2 = 25 \)
• \( b^2 = 12^2 = 144 \)
• \( a^2 + b^2 = 25 + 144 = 169 \)
• \( c^2 = 13^2 = 169 \)
• \( a^2 + b^2 - c^2 = 169 - 169 = 0 \) (so \( = \))
• Type: Right Triangle (Pythagorean theorem holds).

Triangle 2: \( AC = 10 \) ft, \( AB = 18 \) ft, \( BC = 17 \) ft

Let \( a = 10 \), \( b = 17 \), \( c = 18 \) (longest side).

• \( a^2 = 10^2 = 100 \)
• \( b^2 = 17^2 = 289 \)
• \( a^2 + b^2 = 100 + 289 = 389 \)
• \( c^2 = 18^2 = 324 \)
• \( a^2 + b^2 - c^2 = 389 - 324 = 65 \) (so \( > \))
• Type: Acute Triangle (since \( a^2 + b^2 > c^2 \) for the longest side).

Triangle 3: \( AB = 20 \) yd, \( BC = 13 \) yd, \( AC = 11 \) yd

Let \( a = 11 \), \( b = 13 \), \( c = 20 \) (longest side).

• \( a^2 = 11^2 = 121 \)
• \( b^2 = 13^2 = 169 \)
• \( a^2 + b^2 = 121 + 169 = 290 \)
• \( c^2 = 20^2 = 400 \)
• \( a^2 + b^2 - c^2 = 290 - 400 = -110 \) (so \( < \))
• Type: Obtuse Triangle (since \( a^2 + b^2 < c^2 \) for the longest side).

Part B: Triangles with Given Side Lengths (by \( a, b, c \))

Triangle 1: \( a = 6 \) ft, \( b = 4 \) ft, \( c = 9 \) ft

• \( a^2 = 6^2 = 36 \)
• \( b^2 = 4^2 = 16 \)
• \( a^2 + b^2 = 36 + 16 = 52 \)
• \( c^2 = 9^2 = 81 \)
• \( a^2 + b^2 - c^2 = 52 - 81 = -29 \) (so \( < \))
• Type: Obtuse Triangle.

Triangle 2: \( a = 15 \) yd, \( b = 8 \) yd, \( c = 17 \) yd

• \( a^2 = 15^2 = 225 \)
• \( b^2 = 8^2 = 64 \)
• \( a^2 + b^2 = 225 + 64 = 289 \)
• \( c^2 = 17^2 = 289 \)
• \( a^2 + b^2 - c^2 = 289 - 289 = 0 \) (so \( = \))
• Type: Right Triangle.

Triangle 3: \( a = 9 \) in, \( b = 18 \) in, \( c = 19 \) in

• \( a^2 = 9^2 = 81 \)
• \( b^2 = 18^2 = 324 \)
• \( a^2 + b^2 = 81 + 324 = 405 \)
• \( c^2 = 19^2 = 361 \)
• \( a^2 + b^2 - c^2 = 405 - 361 = 44 \) (so \( > \))
• Type: Acute Triangle.

Triangle 4: \( a = 5 \) ft, \( b = 7 \) ft, \( c = 11 \) ft

• \( a^2 = 5^2 = 25 \)
• \( b^2 = 7^2 = 49 \)
• \( a^2 + b^2 = 25 + 49 = 74 \)
• \( c^2 = 11^2 = 121 \)
• \( a^2 + b^2 - c^2 = 74 - 121 = -47 \) (so \( < \))
• Type: Obtuse Triangle.

Final Tables (Filled)

Part A Table:

Triangle	\( a^2 \)	\( b^2 \)	\( a^2 + b^2 \)	\( c^2 \)	\( a^2 + b^2 - c^2 \)	Acute/Obtuse/Right
\( 10, 17, 18 \) ft	100	289	389	324	\( > \)	Acute
\( 11, 13, 20 \) yd	121	169	290	400	\( < \)	Obtuse

Part B Table:

Sides of the Triangle	\( a^2 \)	\( b^2 \)	\( a^2 + b^2 \)	\( c^2 \)	\( a^2 + b^2 - c^2 \)	Acute/Obtuse/Right
\( 15, 8, 17 \) yd	225	64	289	289	\( = \)	Right
\( 9, 18, 19 \) in	81	324	405	361	\( > \)	Acute
\( 5, 7, 11 \) ft	25	49	74	121	\( < \)	Obtuse

These tables use the Pythagorean theorem (\( a^2 + b^2 = c^2 \) for right triangles) and its extensions ( \( a^2 + b^2 > c^2 \) for acute, \( a^2 + b^2 < c^2 \) for obtuse) to classify triangles.