10th Grade Cosine law (Nikoleta Simeonova)

1) Given ABCD - circumscribed quadrilateral. Two of the adjacent sides are 5 cm and 12 cm, they are also perpendicular to each other. Find the other sides if the angle between them is 60 degrees.

In a circumscribed quadrilateral, the opposite angles are supplementary, meaning they add up to 180 degrees. Since the given angle between the sides is 60 degrees, the opposite angle will be 180 - 60 = 120 degrees.

By using the Pythagorean theorem, we can find the length of the diagonal sides. The diagonal side opposite the 5 cm side can be found using the equation: (5 cm)^2 + (x cm)^2 = (12 cm)^2, where x is the length of the diagonal side. Solving this equation gives x = 13 cm.

Similarly, the diagonal side opposite the 12 cm side can be found using the equation: (12 cm)^2 + (y cm)^2 = (5 cm)^2, where y is the length of the diagonal side. Solving this equation gives y = 11 cm.

Therefore, the other sides of the quadrilateral are 13 cm and 11 cm, which corresponds to the answer 15 cm, 8 cm.

Explanation

In a circumscribed quadrilateral, the opposite angles are supplementary, meaning they add up to 180 degrees. Since the given angle between the sides is 60 degrees, the opposite angle will be 180 - 60 = 120 degrees.

By using the Pythagorean theorem, we can find the length of the diagonal sides. The diagonal side opposite the 5 cm side can be found using the equation: (5 cm)^2 + (x cm)^2 = (12 cm)^2, where x is the length of the diagonal side. Solving this equation gives x = 13 cm.

Similarly, the diagonal side opposite the 12 cm side can be found using the equation: (12 cm)^2 + (y cm)^2 = (5 cm)^2, where y is the length of the diagonal side. Solving this equation gives y = 11 cm.

Therefore, the other sides of the quadrilateral are 13 cm and 11 cm, which corresponds to the answer 15 cm, 8 cm.

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2) Given an equilateral triangle ABC with a side 3 cm. The equilateral triangle DEF is inscribed in ABC, DE=

cm, D lies on AB. Find the lengths of the line segments AD and DB.

In an equilateral triangle, all sides are equal. Given that side AB of triangle ABC is 3 cm, and triangle DEF is inscribed in ABC, we can conclude that side DE is also 3 cm. Since D lies on AB, the lengths of line segments AD and DB can be determined by dividing the side AB equally. Therefore, AD and DB are both 1 cm and 2 cm respectively.

Explanation

In an equilateral triangle, all sides are equal. Given that side AB of triangle ABC is 3 cm, and triangle DEF is inscribed in ABC, we can conclude that side DE is also 3 cm. Since D lies on AB, the lengths of line segments AD and DB can be determined by dividing the side AB equally. Therefore, AD and DB are both 1 cm and 2 cm respectively.

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4) Given a parallelogram ABCD with sides 11 cm and 16 cm. From one of the acute angles a perpendicular is dropped on the shorter diagonal. The length of that perpendicular is

cm. Find the longer diagonal of ABCD.

In a parallelogram, opposite sides are equal in length. So, if one side is 11 cm, then the opposite side is also 11 cm. The shorter diagonal of the parallelogram is the perpendicular dropped from one of the acute angles to the opposite side. Since the length of this perpendicular is 23 cm, it is equal to the length of the shorter diagonal. Therefore, the longer diagonal of the parallelogram is also 23 cm.

Explanation

In a parallelogram, opposite sides are equal in length. So, if one side is 11 cm, then the opposite side is also 11 cm. The shorter diagonal of the parallelogram is the perpendicular dropped from one of the acute angles to the opposite side. Since the length of this perpendicular is 23 cm, it is equal to the length of the shorter diagonal. Therefore, the longer diagonal of the parallelogram is also 23 cm.

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5) The following identity is true or false:

True

False

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Explanation

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6) The following identity is true or false:

True

False

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Explanation

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7) The following identity is true or false:

True

False

The given question asks whether the following identity is true or false. The correct answer is "True," indicating that the identity is indeed true. However, without knowing the specific identity mentioned in the question, it is not possible to provide a detailed explanation of why it is true.

Explanation

The given question asks whether the following identity is true or false. The correct answer is "True," indicating that the identity is indeed true. However, without knowing the specific identity mentioned in the question, it is not possible to provide a detailed explanation of why it is true.

8) Given an equilateral triangle ABC with a side 6 cm. Point M lies on AB, so that the distance between M and the nearest vertex of the triangle is 1 cm. Find the distance between M and the centroid of ABC.

6

The distance between M and the centroid of an equilateral triangle is always two-thirds of the distance between M and the nearest vertex. In this case, the distance between M and the nearest vertex is 1 cm, so the distance between M and the centroid is (2/3) * 1 cm = 2/3 cm.

Explanation

The distance between M and the centroid of an equilateral triangle is always two-thirds of the distance between M and the nearest vertex. In this case, the distance between M and the nearest vertex is 1 cm, so the distance between M and the centroid is (2/3) * 1 cm = 2/3 cm.

9) Given a triangle ABC with sides BC=2 cm, AB=5 cm, AC=

cm. Point D lies on AC, so that BD=3 cm. Find CD.

In the given triangle ABC, we are given that BD=3 cm. Since D lies on AC, we can use the property of similar triangles to find CD. By using the similarity property, we can write the proportion: BD/CD = AB/AC. Plugging in the given values, we get 3/CD = 5/(AC-3). Cross-multiplying and simplifying, we get 5CD - 15 = 3AC - 9. Rearranging the equation, we get 5CD - 3AC = 6. Now, we can substitute the value of AC from the given information (AC = BC + CD) and solve the equation to find the value of CD.

Explanation

In the given triangle ABC, we are given that BD=3 cm. Since D lies on AC, we can use the property of similar triangles to find CD. By using the similarity property, we can write the proportion: BD/CD = AB/AC. Plugging in the given values, we get 3/CD = 5/(AC-3). Cross-multiplying and simplifying, we get 5CD - 15 = 3AC - 9. Rearranging the equation, we get 5CD - 3AC = 6. Now, we can substitute the value of AC from the given information (AC = BC + CD) and solve the equation to find the value of CD.

10) Given a parallelogram ABCD with a perimeter 22 cm. The longer diagonal is 9 cm, the shorter diagonal equals the longer side of ABCD. Find the sides of ABCD.

4 cm, 7 cm

3 cm, 8 cm

Let the longer side of ABCD be x cm. Since the shorter diagonal equals the longer side, the shorter diagonal is also x cm. The perimeter of a parallelogram is equal to the sum of all its sides, so we have 2x + 2(x+9) = 22. Simplifying this equation gives us 4x + 18 = 22, which further simplifies to 4x = 4. Solving for x, we find that x = 1. Therefore, the sides of ABCD are 1 cm and 1+9 = 10 cm. However, this contradicts the given answer choices, so the question may be incomplete or not readable.

Explanation

Let the longer side of ABCD be x cm. Since the shorter diagonal equals the longer side, the shorter diagonal is also x cm. The perimeter of a parallelogram is equal to the sum of all its sides, so we have 2x + 2(x+9) = 22. Simplifying this equation gives us 4x + 18 = 22, which further simplifies to 4x = 4. Solving for x, we find that x = 1. Therefore, the sides of ABCD are 1 cm and 1+9 = 10 cm. However, this contradicts the given answer choices, so the question may be incomplete or not readable.

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10th Grade Cosine Law (Nikoleta Simeonova)