To determine whether the cut along the diagonal will result in two equal pieces of cake, we can use the SSS (Side-Side-Side) Congruence Theorem.
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Diagram: Consider the rectangular cake with corners labeled:
- \( I \) (top left corner)
- \( J \) (top right corner)
- \( K \) (bottom right corner)
- \( L \) (bottom left corner)
The dimensions provided are:
- Length \( IJ = 9 \) inches (height)
- Length \( JK = 13 \) inches (width)
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Diagonal Cut: If Sarah suggests cutting the cake along the diagonal \( IK \), we will have two triangles: \(\triangle IJK\) and \(\triangle ILK\).
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Comparing the Triangles:
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For Triangle IJK:
- Side \( IJ = 9 \) inches
- Side \( JK = 13 \) inches
- Side \( IK \) can be found using the Pythagorean theorem: \[ IK = \sqrt{IJ^2 + JK^2} = \sqrt{9^2 + 13^2} = \sqrt{81 + 169} = \sqrt{250} = 5\sqrt{10} , \text{inches} \]
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For Triangle ILK:
- Side \( IL = 9 \) inches (height)
- Side \( LK = 13 \) inches (width)
- Side \( IK \) (the diagonal cut already calculated) is still \( 5\sqrt{10} \) inches.
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SSS Congruence:
- Since both triangles share the same side \( IK \) and have the same lengths \( IJ = IL = 9 \) inches and \( JK = LK = 13 \) inches.
- Therefore, by the SSS Congruence Theorem, we conclude that \(\triangle IJK \cong \triangle ILK\).
Since these two triangles are congruent, the areas will also be equal. This shows that each twin will get exactly the same amount of cake, which confirms to Brock that cutting along the diagonal is indeed a fair way to slice the cake.
Answer to the Multiple-Choice Question: The side that Sarah and Brock would be sharing is Segment IK.
1 answer