Question

drag the tiles to the boxes to form correct pairs. not all tiles will be used. determine each segment length in right triangle abc. image of right triangle abc with right angle at b, angle at c is 45°, angle at a is 45°, d is foot of perpendicular from b to ac, ad=7, ac=14. tiles: 7, 14√2, 14√3, 14, 7√3, 7√2. boxes for bc and bd.

Explanation:

Step1: Analyze triangle \( \triangle BDA \)

Since \( \angle A = 45^{\circ} \) and \( \angle BDA=90^{\circ} \), then \( \angle ABD = 45^{\circ} \). So, \( \triangle BDA \) is a \(45 - 45-90\) triangle. In a \(45 - 45 - 90\) triangle, the legs are equal. Given \(AD = 7\), then \(BD=7\) (because in \(45 - 45-90\) triangle \(a=b\), where \(a\) and \(b\) are the legs).

Step2: Analyze triangle \( \triangle BDC \)

\(CD=14 - 7=7\). Since \( \angle C = 45^{\circ} \) and \( \angle BDC = 90^{\circ} \), \( \triangle BDC \) is also a \(45 - 45-90\) triangle. Using the Pythagorean theorem for \(45 - 45-90\) triangle (\(c=a\sqrt{2}\), where \(c\) is the hypotenuse and \(a\) is the leg). Here \(a = 7\) (for \( \triangle BDC\), \(CD = BD = 7\)), so \(BC=\sqrt{7^{2}+7^{2}}=\sqrt{49 + 49}=\sqrt{98}=7\sqrt{2}\)

Answer:

\(BC:7\sqrt{2}\), \(BD:7\)