Question

19 multiple choice 2 points in △def, which of the following is equal to \frac{5}{12}? 20 multiple choice 2 points which quadrilateral does not always have congruent diagonals? rhombus square isosceles trapezoid rectangle

Explanation:

Question 19

Step1: Recall trigonometric ratios

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)

Step2: Calculate \(\tan D\)

For angle \(D\), the opposite side is \(EF = 5\) and the adjacent side is \(DF=12\). So, \(\tan D=\frac{EF}{DF}=\frac{5}{12}\)

Step3: Calculate \(\sin E\)

For angle \(E\), the opposite side is \(DF = 12\) and the hypotenuse is \(DE = 13\). So, \(\sin E=\frac{DF}{DE}=\frac{12}{13}\)

Step4: Calculate \(\cos D\)

For angle \(D\), the adjacent side is \(DF = 12\) and the hypotenuse is \(DE = 13\). So, \(\cos D=\frac{DF}{DE}=\frac{12}{13}\)

Step5: Calculate \(\sin D\)

For angle \(D\), the opposite side is \(EF = 5\) and the hypotenuse is \(DE = 13\). So, \(\sin D=\frac{EF}{DE}=\frac{5}{13}\)

• A square is a special case of a rectangle and a rhombus. In a square, the diagonals are congruent (\(d_1 = d_2\)), \(d=\sqrt{2}a\) (where \(a\) is the side length)
• In an isosceles trapezoid, the non - parallel sides are equal and the diagonals are congruent. We can prove this using triangle congruence (\(\triangle ABC\cong\triangle BAD\) for isosceles trapezoid \(ABCD\) with \(AD\parallel BC\) and \(AB = CD\))
• In a rectangle, the diagonals are congruent. Using the Pythagorean theorem, if the sides of the rectangle are \(l\) and \(w\), \(d_1=\sqrt{l^{2}+w^{2}}\) and \(d_2=\sqrt{l^{2}+w^{2}}\)
• In a rhombus, the diagonals are perpendicular bisectors of each other, but they are not always congruent. The lengths of the diagonals of a rhombus are \(d_1 = 2a\sin\theta\) and \(d_2=2a\cos\theta\) (where \(a\) is the side length and \(\theta\) is one of the vertex angles), and \(d_1=d_2\) only when \(\theta = 90^{\circ}\) (i.e., when the rhombus is a square)

Answer:

\(\tan D\)

Question 20