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.
A prime number is defined as a number greater than 1 that has exactly two distinct positive divisors: 1 and the number itself. For example, if we consider the prime number 7, its only factors are 1 and 7. Thus, by definition, any prime number will always have exactly two factors: 1 and the number itRead more
A prime number is defined as a number greater than 1 that has exactly two distinct positive divisors: 1 and the number itself. For example, if we consider the prime number 7, its only factors are 1 and 7.
Thus, by definition, any prime number will always have exactly two factors: 1 and the number itself. This makes the total number of factors of a prime number equal to 2.
The other options are incorrect because:
– “1” is incorrect since a prime number has two factors, not one.
– “Zero” is incorrect since every number has at least one factor (itself).
– “3” is incorrect since a prime number cannot have more than two factors.
The HCF of two numbers is 18, and their product is 12960. To find their LCM, we use the relationship between HCF and LCM: HCF × LCM = Product of the two numbers. Substituting the given values: 18 × LCM = 12960. Solving for LCM: LCM = 12960 / 18 LCM = 720. Thus, the LCM of the two numbers is 720. ExpRead more
The HCF of two numbers is 18, and their product is 12960. To find their LCM, we use the relationship between HCF and LCM:
HCF × LCM = Product of the two numbers.
Substituting the given values:
18 × LCM = 12960.
Solving for LCM:
LCM = 12960 / 18
LCM = 720.
Thus, the LCM of the two numbers is 720.
Explanation:
The formula HCF × LCM = Product of the numbers is a fundamental property of HCF and LCM. Since the HCF and the product are given, we can directly calculate the LCM using this formula. The other options (420, 600, and 800) do not satisfy this relationship and are therefore incorrect.
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
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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The total number of factors of a prime number is
A prime number is defined as a number greater than 1 that has exactly two distinct positive divisors: 1 and the number itself. For example, if we consider the prime number 7, its only factors are 1 and 7. Thus, by definition, any prime number will always have exactly two factors: 1 and the number itRead more
A prime number is defined as a number greater than 1 that has exactly two distinct positive divisors: 1 and the number itself. For example, if we consider the prime number 7, its only factors are 1 and 7.
Thus, by definition, any prime number will always have exactly two factors: 1 and the number itself. This makes the total number of factors of a prime number equal to 2.
The other options are incorrect because:
– “1” is incorrect since a prime number has two factors, not one.
– “Zero” is incorrect since every number has at least one factor (itself).
– “3” is incorrect since a prime number cannot have more than two factors.
Thus, the correct answer is 2.
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The HCF of two numbers is 18 and their product is 12960. Their LCM will be
The HCF of two numbers is 18, and their product is 12960. To find their LCM, we use the relationship between HCF and LCM: HCF × LCM = Product of the two numbers. Substituting the given values: 18 × LCM = 12960. Solving for LCM: LCM = 12960 / 18 LCM = 720. Thus, the LCM of the two numbers is 720. ExpRead more
The HCF of two numbers is 18, and their product is 12960. To find their LCM, we use the relationship between HCF and LCM:
HCF × LCM = Product of the two numbers.
Substituting the given values:
18 × LCM = 12960.
Solving for LCM:
LCM = 12960 / 18
LCM = 720.
Thus, the LCM of the two numbers is 720.
Explanation:
The formula HCF × LCM = Product of the numbers is a fundamental property of HCF and LCM. Since the HCF and the product are given, we can directly calculate the LCM using this formula. The other options (420, 600, and 800) do not satisfy this relationship and are therefore incorrect.
Thus, the correct answer is 720.
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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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