ssrs barcode font free Triangles have equal areas if they have congruent bases and congruent altitudes. in Objective-C

Maker QR-Code in Objective-C Triangles have equal areas if they have congruent bases and congruent altitudes.

Triangles have equal areas if they have congruent bases and congruent altitudes.
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Thus in Fig. 9-18, the area of ^CAB equals the area of ^CAD.
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A median divides a triangle into two triangles with equal areas.
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Thus in Fig. 9-19, where BM is a median, the area of ^AMB equals the area of ^BMC since they have congruent bases (AM > MC) and common altitude BD.
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Fig. 9-19
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Triangles are equal in area if they have a common base and their vertices lie on a line parallel to the base.
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Thus in Fig. 9-20, the area of ^ABC is equal to the area of ^ADC.
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SOLVED PROBLEMS
9.9 Proving an equal-areas problem
Given: Trapezoid ABCD (BC i AD) Diagonals AC and BD To Prove: Area(^AEB) area(^DEC) Plan: Use Principle 4 to obtain area(^ABD) Then use the Subtraction Postulate.
area(^ACD).
CHAPTER 9 Areas
PROOF:
Statements 1. BC i AD 2. Area(^ABD) 3. Area(^AED) 4. Area(^AEB) area(^ACD) area(^AED) area(^DEC) 1. Given
Reasons 2. Triangles have equal area if they have a common base and their vertices lie on a line parallel to the base. 3. Identity Postulate 4. Subtraction Postulate
9.10 Proving an equal-areas problem stated in words Prove that if M is the midpoint of diagonal AC in quadrilateral ABCD, and BM and DM are drawn, then the area of quadrilateral ABMD equals the area of quadrilateral CBMD.
Solution
Quadrilateral ABCD M is midpoint of diagonal AC. To Prove: Area of quadrilateral ABMD equals area of quadrilateral CBMD. Plan: Use Principle 3 to obtain two pairs of triangles which are equal in area. Then use the Addition Postulate.
PROOF:
Given:
Statements 1. M is the midpoint of AC. 2. BM is a median of ^ACB. DM is a median of ^ACD. 3. Area (^AMB) area(^BMC), Area(^AMD) area(^DMC). 4. Area of quadrilateral ABMD equals area of quadrilateral CBMD.
Reasons 1. Given 2. A line from a vertex of a triangle to the midpoint of the opposite side is a median. 3. A median divides a triangle into two triangles of equal area. 4. If equals are added to equals, the results are equal.
9.7 Comparing Areas of Similar Polygons
The areas of similar polygons are to each other as the squares of any two corresponding segments. Thus if ^ABC ~ ^A B C and the area of ^ABC is 25 times the area of ^A B C , then the ratio of the lengths of any two corresponding sides, medians, altitudes, radii of inscribed or circumscribed circles, and such is 5:1.
SOLVED PROBLEMS
9.11 Ratio of areas and segments of similar triangles Find the ratio of the areas of two similar triangles (a) if the ratio of the lengths of two corresponding sides is 3:5; (b) if their perimeters are 12 and 7. Find the ratio of the lengths of a pair of (c) corresponding sides if the ratio of the areas is 4:9; (d ) corresponding medians if the areas are 250 and 10.
CHAPTER 9 Areas
Solutions
(a) A A (b) A A Q ss R
3 2 Q R 5 Q 12 2 R 7
9 25 144 49
(c) Q s R s
A A A A
s 4 or 9 s 250 m or m 10
2 3 5
p 2 Qp R
2 (d) Q m R m
9.12 Proportions derived from similar polygons (a) The areas of two similar polygons are 80 and 5. If a side of the smaller polygon has length 2, find the length of the corresponding side of the larger polygon. (b) The corresponding diagonals of two similar polygons have lengths 4 and 5. If the area of the larger polygon is 75, find the area of the smaller polygon.
Solutions
2 (a) Q s R s
A so s 2 A , Q2R
2 Q d R , so A d 75
80 5
16, Then
4 and s
(b) A A
4 2 Q R . Then A 75 Q 16 R 5 25
SUPPLEMENTARY PROBLEMS
9.1. Find the area of a rectangle (a) If the base has length 11 in and the altitude has length 9 in (b) If the base has length 2 ft and the altitude has length 1 ft 6 in (c) If the base has length 25 and the perimeter is 90 (d) If the base has length 15 and the diagonal has length 17 (e) If the diagonal has length 12 and the angle between the diagonal and the base measures 60 (f) If the diagonal has length 20 and the angle between the diagonal and the base measures 30 (g) If the diagonal has length 25 and the lengths of the sides are in the ratio of 3:4 (h) If the perimeter is 50 and the lengths of the sides are in the ratio of 2:3 9.2. Find the area of a rectangle inscribed in a circle (a) If the radius of the circle is 5 and the base has length 6 (b) If the radius of the circle is 15 and the altitude has length 24 (c) If the radius and the altitude both have length 5 (d) If the diameter has length 26 and the base and altitude are in the ratio of 5:12 9.3. Find the base and altitude of a rectangle (a) If its area is 28 and the base has a length of 3 more than the altitude (b) If its area is 72 and the base is twice the altitude (9.1) (9.1) (9.1)
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