In an isosceles triangle ABC, given that AB = AC = 25 cm and BC = 14 cm, we need to find the measure of the altitude from A to BC. Let the altitude from A meet BC at point D. Since the triangle is isosceles, the altitude AD also acts as the median, dividing BC into two equal parts: BD = DC = BC / 2Read more
In an isosceles triangle ABC, given that AB = AC = 25 cm and BC = 14 cm, we need to find the measure of the altitude from A to BC.
Let the altitude from A meet BC at point D. Since the triangle is isosceles, the altitude AD also acts as the median, dividing BC into two equal parts:
BD = DC = BC / 2 = 14 / 2 = 7 cm.
Now, consider ΔABD, which is a right triangle because AD is perpendicular to BC. Using the Pythagorean theorem:
AB² = AD² + BD²
25² = AD² + 7²
625 = AD² + 49
AD² = 625 – 49
AD² = 576
AD = √576 = 24 cm.
Thus, the measure of the altitude from A on BC is 24 cm.
When AD is the altitude from vertex A to side BC, it divides ΔABC into two right triangles, ΔABD and ΔACD. Using the property of the area of a triangle, we know that the area can be expressed in two ways: 1. Area = (1/2) × base × height = (1/2) × BC × AD. 2. Area = (1/2) × AB × AC × sin(∠BAC). EquatRead more
When AD is the altitude from vertex A to side BC, it divides ΔABC into two right triangles, ΔABD and ΔACD. Using the property of the area of a triangle, we know that the area can be expressed in two ways:
1. Area = (1/2) × base × height = (1/2) × BC × AD.
2. Area = (1/2) × AB × AC × sin(∠BAC).
Equating the two expressions for the area:
(1/2) × BC × AD = (1/2) × AB × AC × sin(∠BAC).
Since sin(∠BAC) = AD / AB (from the definition of sine in ΔABD), substituting this value simplifies the equation to:
BC × AD = AB × AC.
In ΔABC, given that ∠B = 90°, AB = 6 cm, and BC = 8 cm, we can determine sin A using the definition of sine in a right triangle. First, calculate the hypotenuse AC using the Pythagorean theorem: AC² = AB² + BC² AC² = 6² + 8² AC² = 36 + 64 AC² = 100 AC = √100 = 10 cm Now, recall that sin A is definedRead more
In ΔABC, given that ∠B = 90°, AB = 6 cm, and BC = 8 cm, we can determine sin A using the definition of sine in a right triangle.
First, calculate the hypotenuse AC using the Pythagorean theorem:
AC² = AB² + BC²
AC² = 6² + 8²
AC² = 36 + 64
AC² = 100
AC = √100 = 10 cm
Now, recall that sin A is defined as the ratio of the length of the side opposite to ∠A (BC) to the hypotenuse (AC):
sin A = (opposite side) / (hypotenuse)
sin A = BC / AC
sin A = 8 / 10
Similarity of triangles is based on the principle that when two triangles have the same shape but not necessarily the same size, they are considered similar. This occurs if and only if their corresponding angles are equal. When the angles are equal, the ratios of the lengths of their corresponding sRead more
Similarity of triangles is based on the principle that when two triangles have the same shape but not necessarily the same size, they are considered similar. This occurs if and only if their corresponding angles are equal. When the angles are equal, the ratios of the lengths of their corresponding sides are also equal (this is known as the Angle-Angle or AA criterion for similarity).
The other options are incorrect because:
– “Their corresponding sides are equal” describes congruence, not similarity.
– “Their perimeters are equal” does not guarantee similarity, as triangles with the same perimeter can have different shapes.
– “Their areas are equal” also does not ensure similarity, as triangles with the same area can have different shapes and dimensions.
Thus, the correct answer is “Their corresponding angles are equal.”
In an isosceles triangle ABC, if AB = AC = 25 cm and BC = 14 cm, then the measure of altitude from A on BC is
In an isosceles triangle ABC, given that AB = AC = 25 cm and BC = 14 cm, we need to find the measure of the altitude from A to BC. Let the altitude from A meet BC at point D. Since the triangle is isosceles, the altitude AD also acts as the median, dividing BC into two equal parts: BD = DC = BC / 2Read more
In an isosceles triangle ABC, given that AB = AC = 25 cm and BC = 14 cm, we need to find the measure of the altitude from A to BC.
Let the altitude from A meet BC at point D. Since the triangle is isosceles, the altitude AD also acts as the median, dividing BC into two equal parts:
BD = DC = BC / 2 = 14 / 2 = 7 cm.
Now, consider ΔABD, which is a right triangle because AD is perpendicular to BC. Using the Pythagorean theorem:
AB² = AD² + BD²
25² = AD² + 7²
625 = AD² + 49
AD² = 625 – 49
AD² = 576
AD = √576 = 24 cm.
Thus, the measure of the altitude from A on BC is 24 cm.
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In ΔABC, if AD ⊥ BC, then which of these is always true?
When AD is the altitude from vertex A to side BC, it divides ΔABC into two right triangles, ΔABD and ΔACD. Using the property of the area of a triangle, we know that the area can be expressed in two ways: 1. Area = (1/2) × base × height = (1/2) × BC × AD. 2. Area = (1/2) × AB × AC × sin(∠BAC). EquatRead more
When AD is the altitude from vertex A to side BC, it divides ΔABC into two right triangles, ΔABD and ΔACD. Using the property of the area of a triangle, we know that the area can be expressed in two ways:
1. Area = (1/2) × base × height = (1/2) × BC × AD.
2. Area = (1/2) × AB × AC × sin(∠BAC).
Equating the two expressions for the area:
(1/2) × BC × AD = (1/2) × AB × AC × sin(∠BAC).
Since sin(∠BAC) = AD / AB (from the definition of sine in ΔABD), substituting this value simplifies the equation to:
BC × AD = AB × AC.
Thus, the correct answer is AB × AC = BC × AD.
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In ΔABC, if ∠B = 90°, AB = 6 cm, and BC = 8 cm, then sin A equals:
In ΔABC, given that ∠B = 90°, AB = 6 cm, and BC = 8 cm, we can determine sin A using the definition of sine in a right triangle. First, calculate the hypotenuse AC using the Pythagorean theorem: AC² = AB² + BC² AC² = 6² + 8² AC² = 36 + 64 AC² = 100 AC = √100 = 10 cm Now, recall that sin A is definedRead more
In ΔABC, given that ∠B = 90°, AB = 6 cm, and BC = 8 cm, we can determine sin A using the definition of sine in a right triangle.
First, calculate the hypotenuse AC using the Pythagorean theorem:
AC² = AB² + BC²
AC² = 6² + 8²
AC² = 36 + 64
AC² = 100
AC = √100 = 10 cm
Now, recall that sin A is defined as the ratio of the length of the side opposite to ∠A (BC) to the hypotenuse (AC):
sin A = (opposite side) / (hypotenuse)
sin A = BC / AC
sin A = 8 / 10
Thus, the correct answer is 8/10.
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Two triangles are similar if:
Similarity of triangles is based on the principle that when two triangles have the same shape but not necessarily the same size, they are considered similar. This occurs if and only if their corresponding angles are equal. When the angles are equal, the ratios of the lengths of their corresponding sRead more
Similarity of triangles is based on the principle that when two triangles have the same shape but not necessarily the same size, they are considered similar. This occurs if and only if their corresponding angles are equal. When the angles are equal, the ratios of the lengths of their corresponding sides are also equal (this is known as the Angle-Angle or AA criterion for similarity).
The other options are incorrect because:
– “Their corresponding sides are equal” describes congruence, not similarity.
– “Their perimeters are equal” does not guarantee similarity, as triangles with the same perimeter can have different shapes.
– “Their areas are equal” also does not ensure similarity, as triangles with the same area can have different shapes and dimensions.
Thus, the correct answer is “Their corresponding angles are equal.”
Click here for more:
See lesshttps://www.tiwariacademy.in/ncert-solutions-class-10-maths-chapter-6/