Can you judge by just looking?

Geometry Level 1

Which triangle has a larger area?

The areas are equal. Yellow Triangle Red Triangle

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3 solutions

Syed Hamza Khalid
Nov 11, 2017

Area of Yellow triangle b a s e × h e i g h t 2 = 80 × h e i g h t 2 where height = 4 1 2 4 0 2 (By the Pythagorean theorem) 9 Area of the yellow triangle 80 × 9 2 = 360 \text{Area of Yellow triangle} \to \frac{base \times height}{2} = \frac{80 \times height}{2} \\ \text{where height =} \sqrt{41^2 - 40^2} \color{#3D99F6} \text{ (By the Pythagorean theorem)} \color{#20A900} \to 9 \\ \therefore \text{Area of the yellow triangle } \to \frac{80 \times 9}{2} \\ \large \color{magenta}= 360

Area of Red triangle b a s e × h e i g h t 2 = 18 × h e i g h t 2 where height = 4 1 2 9 2 (By the Pythagorean theorem) 40 Area of the red triangle 18 × 40 2 = 360 \text{Area of Red triangle} \to \frac{base \times height}{2} = \frac{18 \times height}{2} \\ \text{where height =} \sqrt{41^2 - 9^2} \color{#3D99F6} \text{ (By the Pythagorean theorem)} \color{#20A900} \to 40 \\ \therefore \text{Area of the red triangle } \to \frac{18 \times 40}{2} \\ \large \color{magenta}= 360

Hence, the 2 areas are EQUAL. \large \text{Hence, the 2 areas are} \color{#EC7300} \text{ EQUAL. }

Consider the yellow triangle:

s = 41 + 41 + 80 2 = 81 s=\dfrac{41+41+80}{2}=81

A y e l l o w = 81 ( 81 41 ) ( 81 41 ) ( 81 80 ) = 360 A_{yellow}=\sqrt{81(81-41)(81-41)(81-80)}=360

Consider the red triangle:

s = 41 + 41 + 18 2 = 50 s=\dfrac{41+41+18}{2}=50

A r e d = 50 ( 50 41 ) ( 50 41 ) ( 50 18 ) = 360 A_{red}=\sqrt{50(50-41)(50-41)(50-18)}=360

Hence, the areas are equal.

Saksham Jain
Nov 11, 2017

heron s formula as sides are equal areas are equal

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