TWO TRIANGLES HAVING THE SAME BASE AND EQUAL AREAS LIE BETWEEN THE SAME PARALLELS
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The document provides two examples of geometric proofs involving areas of shapes. The first example proves that a triangle's median divides it into two equal areas. It does so by drawing an altitude and showing the two resulting triangles have the same base and altitude. The second example proves the area of a quadrilateral ABCD equals the area of triangle ADE by showing two triangles on the same base and between parallel lines have equal areas.
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1. Definitions of parallel lines, congruent triangles, and similar triangles.
2. Examples of proofs of the Triangle Midpoint Theorem - which states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.
3. An example proof of the Triangle Intercept Theorem - which states that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.
Congruent Triangles
The document discusses different theorems for proving triangles are congruent:
- Side-Side-Side (SSS) - If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
- Side-Angle-Side (SAS) - If two sides and the included angle of one triangle are congruent to those of another, the triangles are congruent.
- Angle-Side-Angle (ASA) - If two angles and the included side of one triangle are congruent to those of another, the triangles are congruent.
- Angle-Angle-Side (AAS) - If two angles and the non-included side of one triangle are congr
2sol
1) The document contains 6 multi-step math problems and their solutions.
2) The problems involve concepts like trigonometry, inequalities, divisibility, and proportions in geometric figures.
3) The solutions use techniques like cyclic quadrilaterals, similarity of triangles, and algebraic manipulation of equations.
Circlestangentchordtheorem
This document discusses key concepts in circles such as chords, radii, diameters, tangents, and secants. It presents several important theorems: the Tangent-Chord Theorem states that the angle between a tangent and chord is equal to the inscribed angle on the other side of the chord. The Intersecting Chord Theorem relates the lengths of segments formed by two intersecting chords. The Tangent-Secant Theorem equates the product of a secant and its external segment to the square of the tangent. Examples are provided to demonstrate applications of these theorems.
Midpoit Theorem.pdf
1) The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half its length. This is known as the midpoint theorem.
2) The converse of the midpoint theorem states that a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side.
3) The theorem on intercepts states that if a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them also makes equal intercepts.
Presentation1
A quadrilateral is a shape with four sides, four angles, and four vertices. There are six types of quadrilaterals: trapezium, parallelogram, rectangle, rhombus, square, and kite. A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. The diagonals of a parallelogram bisect each other and divide the parallelogram into two congruent triangles. Opposite sides and opposite angles of a parallelogram are equal.
Q. In an isosceles triangle ABC with AB = AC, D and E are points on BC such t...
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Angles opposite to equal sides of an isosceles triangle are equal.
Q. In an isosceles triangle ABC with AB = AC, D and E are points on BC such that BE = CD. Show that AD = AE.
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TWO TRIANGLES HAVING THE SAME BASE AND EQUAL AREAS LIE BETWEEN THE SAME PARALLELS
- 2. EXAMPLE- 1
- 3. Show that a median of a triangle divides it into two triangles of equal areas.
- 5. Let ABC be a triangle and let AD be one of its medians suppose, you wish to show area(ABD) = area(ACD) Since the formula for area involves altitude ,let us draw AN┴BC. Now area(ABD) = 1 ∕ 2 x base x altitude(of ∆ADB) = 1 ∕ 2 x BD x AN = 1 ∕ 2 x CD x AN(As BD = CD) = 1 ∕ 2 x base x altitude(of ∆ACD) = area(ACD)
- 6. EXAMPLE- 2
- 7. ABCD is a quadrilateral and BE||AC and also BE meets DC produced at E. Show that area of ∆ADE is equal to the area of the quadrilateral ABCD.
- 9. Solution: In the given figure ∆BAC and ∆EAC lie on the same base AC and between the same parallels AC and BE. Therefore, area(BAC) = area(EAC)(two triangles on the same base and between the same parallels are equal in area) So, area(BAC) +area(ADC) = area(EAC) + area(ADC) (adding same areas on both sides) area(ABCD) = area(ADE)