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In ΔABC, AB = BC and m∠ABC = 90°. D and E are the midpoints of AB and BC, respectively.
If the length of CA is 9 units, the length of DE is _____ units and m∠CAB is _____ °.
In triangle ΔABC, since AB = BC and m∠ABC = 90°, it is an isosceles right triangle. Let’s denote the lengths of AB and BC as x.
Using the Pythagorean theorem, we know:
[
CA^2 = AB^2 + BC^2
]
[
9^2 = x^2 + x^2
]
[
81 = 2x^2
]
[
x^2 = frac{81}{2}
]
[
x = frac{9}{sqrt{2}} = frac{9sqrt{2}}{2} text{ (after rationalizing)}
]
Next, to find the length of DE, which is the segment connecting midpoints D and E, we use the midpoint theorem.
DE is parallel to AC and half its length:
[
DE = frac{1}{2} times CA = frac{1}{2} times 9 = 4.5
]
So, the length of DE is 4.5 units.
Now, to find m∠CAB, since it is a right triangle with equal legs, it follows that:
[
m∠CAB = 45°
]
Therefore, the answers are:
Length of DE is 4.5 units, and m∠CAB is 45 °.