Browse Congruence & similarity

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Congruent triangles & proof

16 questions

LessonNot started
Triangle ABCABC has AB=8AB = 8 cm and BC=5BC = 5 cm, and the angle at AA (which is not between those two sides) is 30°30°.

(a) Show that there are two different triangles that fit this description, by working out the two possible sizes of angle ACBACB. Give your answers correct to 1 decimal place.

(b) Explain what this shows about "SSA" as a condition for congruence.
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ABCABC is an isosceles triangle with AB=ACAB = AC. MM is the midpoint of BCBC.

(a) Prove that triangle ABMABM is congruent to triangle ACMACM.

(b) Hence prove that AMAM is perpendicular to BCBC.
●●●●●Level 56 marksStart
AA and BB are points on a circle with centre OO. MM is the midpoint of the chord ABAB.OABM(a) Prove that triangle OAMOAM is congruent to triangle OBMOBM.

(b) Hence explain why OMOM is perpendicular to ABAB.
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ABCABC is an isosceles triangle with AB=ACAB = AC.

DD is a point on ABAB and EE is a point on ACAC such that BD=CEBD = CE.

(a) Prove that triangle DBCDBC is congruent to triangle ECBECB.

(b) Hence show that DC=EBDC = EB.
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RR and SS are points on a circle, centre OO.

The tangent to the circle at RR and the tangent to the circle at SS meet at the point TT.ORSTProve that RT=STRT = ST

(4 marks)
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Similar shapes & missing lengths

28 questions

LessonNot started
A solid cone has base radius 1212 cm and perpendicular height 2020 cm.

The top of the cone is cut off by a cut parallel to the base, removing a smaller cone of perpendicular height 55 cm. The solid that is left is called a frustum.20 cm12 cm5 cmDiagram NOT accurately drawn

Volume of a cone =13πr2h= \frac{1}{3}\pi r^2 h

Work out the volume of the frustum. Give your answer in terms of π\pi.
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Two solid statues are mathematically similar.

The smaller statue has volume 128128 cm³ and surface area 9696 cm².

The larger statue has volume 250250 cm³.

Work out the surface area of the larger statue.
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Two mathematically similar vases have surface areas 3636 cm² and 8181 cm².

The larger vase has volume 540540 cm³.

Work out the volume of the smaller vase.
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XX, YY and ZZ are three mathematically similar solid square-based pyramids made from the same material.

XX has a mass of 5454 g

YY has a mass of 250250 g

YY has a total surface area of 400400 cm²

ZZ has a total surface area of 196196 cm²

Work out

height of pyramid XX : height of pyramid YY : height of pyramid ZZ

Give your answer in its simplest form.

(4 marks)
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The diagram shows three triangles, OABOAB, OCDOCD and OEFOEF.OACEBDFOACEOACE and OBDFOBDF are straight lines.

ABAB, CDCD and EFEF are parallel.

OA:AC:CE=2:3:4OA : AC : CE = 2 : 3 : 4

Show that

area of triangle OABOAB : area of ABDCABDC : area of CDFECDFE =4:21:56= 4 : 21 : 56

(3 marks)
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FGHFGH and JFGJFG are similar isosceles triangles.FGHJFG=FHFG = FH

JF=JGJF = JG

GHJGHJ is a straight line.

GH:HJ=9:7GH : HJ = 9 : 7

Find the ratio FG:FJFG : FJ

(3 marks)
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A solid frustum is made by removing a small cone from the top of a large cone.15 cm6 cm6.5 cmCurved surface area of cone =πrl= \pi r l

The large cone has a slant height of 1515 cm.

The base of the large cone has a radius of 6.56.5 cm.

The slant height of the frustum is 66 cm.

Calculate the total surface area of the frustum.

Give your answer correct to 33 significant figures.

(5 marks)
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