1.An isosceles triangle with each leg measuring 13 cm is inscribed in a circle . if the altitude to the base is 12 cm find the radius of the circle
2. Circles a and b are tangent at point c. p is on circle a and q is on circle b such that pq is tangent to both circles. Given ac= 3 cm and bc= 8 cm, find pq
3.Ab is a chord of a circle with center o and radius 52 cm . point m divides the chord ab such that am = 63 cm and mb=33 cm find om
4. A circle is inscribed in a triangle whose sides are 10, 10 and 12 units . a second smaller circle is inscribed tangent to the first circle and to the equal sides of the triangle. Find the radius of the second triangle.
1. In an isosceles triangle, the altitude from the vertex angle bisects the base. Therefore, the base of the triangle is divided into two congruent segments of length 6 cm each.
2. Let's denote the points of tangency between pq and circles a and b as X and Y, respectively. Due to the properties of tangents, we have ∆ACX ~ ∆BCY (by AA similarity). This means that their corresponding sides are proportional.
Using the given information, we can set up the following proportion: AC/BC = PX/PY. Plugging in the values, we get 3/8 = PX/PY.
Since the lengths of AC and BC are given, we can calculate PX and PY using the proportion. PX = (3/8) * PQ and PY = (8/3) * PQ.
3. The line segment OM is the perpendicular bisector of the chord AB. The perpendicular bisector of a chord always passes through the center of the circle.
Therefore, OM is the radius of the circle. To find its length, we can use the Pythagorean theorem in triangle OAM: OM^2 = OA^2 - AM^2.
OA is the radius of the circle (52 cm), and AM is half of AB (63 cm / 2 = 31.5 cm).
Substituting the values, we get OM^2 = (52 cm)^2 - (31.5 cm)^2. Solving this equation will give us OM^2, and taking the square root will give us OM.
4. Let's denote the radius of the first (larger) circle as R and the radius of the second (smaller) circle as r.
The inradius of a triangle can be calculated using the formula: inradius = area / semiperimeter, where the area of the triangle can be calculated using Heron's formula.
The semiperimeter of the triangle is (10 + 10 + 12) / 2 = 16.
Using Heron's formula, we can calculate the area: area = sqrt(s * (s - a) * (s - b) * (s - c)), where a, b, and c are the side lengths of the triangle.
Plugging in the values, we get area = sqrt(16 * (16 - 10) * (16 - 10) * (16 - 12)) = sqrt(16 * 6 * 6 * 4) = 48.
Now, let's calculate the inradius: inradius = area / semiperimeter = 48 / 16 = 3.
Since the second (smaller) circle is inscribed tangent to the sides of the triangle, its radius is equal to the inradius, which is 3 units.
1. To find the radius of the circle, we need to use the properties of an isosceles triangle inscribed in a circle. Here's how you can do it:
Step 1: Draw the isosceles triangle ABC with each leg measuring 13 cm, and let O be the center of the circle.
Step 2: Draw the altitude CD to the base AB, where D is the foot of the altitude. Since it is an isosceles triangle, the altitude will divide the base into two equal parts.
Step 3: Since AD = DB = 6.5 cm (half of 13 cm), we have a right-angled triangle ADC with AD = 6.5 cm and CD = 12 cm.
Step 4: Using the Pythagorean theorem, we can find the length of AO, which is the radius of the circle. It can be calculated as follows:
AO^2 = AC^2 - OC^2
AO^2 = (AD^2 + CD^2) - OC^2
AO^2 = (6.5^2 + 12^2) - OC^2
AO^2 = 42.25 + 144 - OC^2
AO^2 = 186.25 - OC^2
OC^2 = 186.25 - AO^2
Step 5: Since OC is also equal to the radius of the circle, we can substitute it back into the equation:
OC^2 = 186.25 - AO^2
OC^2 = 186.25 - (OC^2) (Substituting AO^2 with OC^2)
2OC^2 = 186.25
OC^2 = 93.125
OC ≈ √93.125
OC ≈ 9.644 cm
So, the radius of the circle is approximately 9.644 cm.
2. To find the length of pq, we can use the property that tangents to a circle from a common external point are equal in length. Here's how you can do it:
Step 1: Draw circles A and B tangent at point C. Draw the tangent pq to both circles.
Step 2: Label the intersection points of pq with circles A and B as P and Q, respectively.
Step 3: Since tangents from a point to a circle are equal, we have AC = AP and BC = BQ.
Step 4: Therefore, pq = AP + BQ = AC + BC = 3 cm + 8 cm = 11 cm.
So, the length of pq is 11 cm.
3. To find the length of OM, we can use the property that the perpendicular bisector of a chord passes through the center of a circle. Here's how you can do it:
Step 1: Draw circle O with chord AB dividing it into two parts.
Step 2: Draw the perpendicular bisector of chord AB and label the intersection point with AB as M.
Step 3: Since OM is the perpendicular bisector of AB, AM = MB.
Step 4: We are given that AM = 63 cm and MB = 33 cm. Since AM = MB, their lengths should add up to the length of the chord:
AM + MB = 63 cm + 33 cm
2 AM = 96 cm
AM = 48 cm
Step 5: OM is the radius of the circle, and since OM bisects the chord, we have OM = radius = 52 cm.
So, the length of OM (the radius of the circle) is 52 cm.
4. To find the radius of the second circle inscribed in the triangle, we can use the formulas for the radius of an inscribed circle in a triangle and the length of the triangle's sides. Here's how you can do it:
Step 1: Draw the triangle with sides 10, 10, and 12 units, and let r be the radius of the second circle.
Step 2: Let A, B, and C be the vertices of the triangle, and let D, E, and F be the points where the incircle (second circle) is tangent to sides BC, AC, and AB, respectively.
Step 3: By the formula for the radius of an inscribed circle, we know that the area of the triangle is equal to the semi-perimeter multiplied by the radius:
Area of triangle ABC = Semiperimeter * r
Step 4: The semi-perimeter of the triangle ABC can be calculated as:
Semi-perimeter = (10 + 10 + 12)/2 = 16
Step 5: The area of the triangle ABC can be calculated using Heron's formula:
Area = √(s * (s - a) * (s - b) * (s - c))
where s is the semi-perimeter, and a, b, and c are the lengths of the triangle's sides.
Area = √(16 * (16 - 10) * (16 - 10) * (16 - 12))
Area = √(16 * 6 * 6 * 4)
Area = √(2304)
Area = 48 units^2
Step 6: Substitute the values into the equation for the area of the triangle:
48 = 16 * r
Step 7: Solve for r:
48/16 = r
r = 3 units
So, the radius of the second circle is 3 units.