Question

as shown in the figure, quadrilateral abcd is a square. extend ab until reaching point e such that ae = ac. then the measure of ∠bce is ____. a. 67.5° b. 22.5° c. 30° d. 45° as shown in the figure, in square abcd, e is a point on cd. if be = 3, and ce = 1, then the length of the diagonal of square abcd is ____. a. 8 b. 4√2 c. 6 d. 4

Explanation:

Step1: Find angle BAC in square ABCD

In square \(ABCD\), the diagonal \(AC\) bisects the angle \(BAD\). Since \(\angle BAD = 90^{\circ}\), then \(\angle BAC=45^{\circ}\).

Step2: Find angle ACE in \(\triangle ACE\)

Given \(AE = AC\), so \(\triangle ACE\) is isosceles. Then \(\angle ACE=\angle AEC\). Using the angle - sum property of a triangle in \(\triangle ACE\), \(\angle ACE=\angle AEC=\frac{180^{\circ}-\angle BAC}{2}=\frac{180 - 45}{2}=67.5^{\circ}\).

Step3: Calculate angle BCE

\(\angle BCE=\angle ACE-\angle ACB\). In square \(ABCD\), \(\angle ACB = 45^{\circ}\). So \(\angle BCE=67.5^{\circ}-45^{\circ}=22.5^{\circ}\).

for second - part:

Step1: Find side length of square using Pythagorean theorem

In right - triangle \(BCE\), let the side length of the square be \(x\). By the Pythagorean theorem in \(\triangle BCE\), \(BE^{2}=BC^{2}+CE^{2}\). Given \(BE = 3\) and \(CE = 1\), then \(3^{2}=x^{2}+1^{2}\), so \(x^{2}=9 - 1=8\), and \(x = 2\sqrt{2}\).

Step2: Calculate the length of the diagonal of the square

The length of the diagonal \(d\) of a square with side length \(x\) is \(d=\sqrt{2}x\). Substituting \(x = 2\sqrt{2}\), we get \(d=\sqrt{2}\times2\sqrt{2}=4\).

Answer:

B. \(22.5^{\circ}\)