Question

drag each length to the correct location on the image. each length can be used more than once, but not all lengths will be used. what are the missing segment lengths shown in the image? 10 20√3 10√2 10√3 20 20√2

Explanation:

Step1: Analyze right - triangle ADC

In right - triangle \(ADC\) with \(\angle D = 45^{\circ}\) and \(\angle C=90^{\circ}\), it is a \(45 - 45-90\) triangle. The ratio of the sides of a \(45 - 45-90\) triangle is \(1:1:\sqrt{2}\). Given the hypotenuse \(AD = 20\), if the legs are \(x\), then by the Pythagorean theorem \(x^{2}+x^{2}=20^{2}\), or \(2x^{2}=400\), \(x^{2} = 200\), \(x = 10\sqrt{2}\). So \(AC=CD = 10\sqrt{2}\).

Step2: Analyze right - triangle ABC

In right - triangle \(ABC\), \(\angle BCA = 45^{\circ}\) and \(\angle ABC=90^{\circ}\), it is also a \(45 - 45-90\) triangle. Since \(AC = 10\sqrt{2}\), if the legs are \(y\), then \(y^{2}+y^{2}=(10\sqrt{2})^{2}\), \(2y^{2}=200\), \(y = 10\). So \(AB = BC=10\).

Answer:

\(AB = 10\), \(BC = 10\), \(AC = 10\sqrt{2}\), \(CD = 10\sqrt{2}\)